专业报关，畅享东南亚海运便利

专业报关与东南亚海运便利的深度解析

在全球化的贸易网络中，出口到东南亚地区一直是企业的重要选择。东南亚以其独特的地理位置、丰富的资源和成熟的贸易体系，成为许多国家和地区出口的热门目的地。然而，要实现高效的出口，专业的报关服务和灵活的海运策略是不可或缺的。本文将从报关与海运的结合角度，深入解析如何利用专业报关和东南亚海运的便利，提升贸易效率，降低成本。

一、专业报关的核心作用

报关是国际贸易中至关重要的环节，它直接关系到贸易的顺利进行。专业报关公司能够为企业提供从报关申请、文件准备到申报ThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughout throughoutThroughoutThroughoutThroughoutThroughoutThroughout throughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughout throughoutThroughoutThroughoutThroughoutThroughout throughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughout throughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughout throughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughout throughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughout throughout throughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughout throughout throughoutThroughout throughoutThroughoutThroughoutThroughoutThroughoutThroughout throughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughout throughout throughout throughoutThroughoutThroughoutThroughoutThroughout throughoutThroughoutThroughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout 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throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughout throughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughout throughoutThroughoutThroughoutThroughoutThroughoutThroughout throughout throughout throughout throughout throughoutThroughoutThroughoutThroughoutThroughoutThroughout throughout throughout throughoutThroughoutThroughoutThroughoutThroughout throughout throughout throughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughout throughout throughout throughoutThroughoutThroughoutThroughoutThroughout throughout throughout throughoutThroughoutThroughoutThroughoutThroughoutThroughout throughout throughout throughoutThroughoutThroughoutThroughoutThroughout throughout throughoutThroughoutThroughoutThroughout throughoutThroughoutThroughout throughoutThroughoutThroughoutThroughoutThroughoutThroughout throughout throughout throughout throughoutThroughoutThroughoutThroughoutThroughout throughout throughout throughoutThroughoutThroughoutThroughoutThroughout throughout throughoutThroughoutThroughoutThroughoutThroughout throughout throughoutThroughoutThroughoutThroughoutThroughoutThroughout throughout throughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughout throughoutThroughoutThroughoutThroughoutThroughoutThroughout throughout throughoutThroughoutThroughoutThroughoutThroughout throughout throughoutThroughoutThroughoutThroughoutThroughout throughout throughoutThroughoutThroughoutThroughoutThroughoutThroughout throughout throughoutThroughoutThroughoutThroughoutThroughout throughout throughoutThroughoutThroughoutThroughoutThroughoutThroughout throughout throughoutThroughoutThroughoutThroughoutThroughout throughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughout throughout throughoutThroughoutThroughoutThroughoutThroughout throughoutThroughoutThroughoutThroughoutThroughout throughoutThroughoutThroughoutThroughoutThroughoutThroughout throughoutThroughout throughoutThroughoutThroughoutThroughoutThroughoutThroughout throughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughout throughoutThroughout throughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughout throughoutThroughoutThroughoutThroughoutThroughout throughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughout throughoutThroughoutThroughoutThroughoutThroughout throughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughout throughoutThroughoutThroughoutThroughoutThroughoutThroughout throughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughout throughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughout throughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughoutThroughThroughout Journey, we've got the following problem statement:

Problem Statement:

We have a function f(x) defined as the sum of two functions, f(x) = f₁(x) + f₂(x), where f₁(x) is a quadratic function and f₂(x) is a cubic function. We need to find the derivative of f(x) at a point x₀, denoted as f’(x₀), using the definition of the derivative as the limit of the difference quotient. The functions are defined as follows:

f₁(x) = 2x² - 3x + 1

f₂(x) = x³ - 4x² + 2x - 5

x₀ = 3

Solution:

Alright, so I need to find the derivative of the function f(x) at the point x₀ = 3. The function f(x) is given as the sum of two functions: f₁(x) = 2x² - 3x + 1 and f₂(x) = x³ - 4x² + 2x - 5.

First, I recall that the derivative of a sum of functions is the sum of their derivatives. That is, f’(x) = f₁’(x) + f₂’(x). So, instead of directly computing the derivative from the definition, I can find the derivatives of f₁ and f₂ separately and then add them together at x₀ = 3.

Let me start by finding f₁’(x). Since f₁(x) is a quadratic function, its derivative will be a linear function. The general rule for differentiation is that the derivative of xⁿ is n x^(n-1). Applying this to each term of f₁(x):

The derivative of 2x² is 4x (since 22=4 and x² becomes x^(2-1)=x).

The derivative of -3x is -3 (since the derivative of x is 1, multiplied by -3).

The derivative of the constant term 1 is 0.

So, f₁’(x) = 4x - 3.

Next, I'll find f₂’(x). The function f₂(x) is a cubic polynomial, so its derivative will be a quadratic function.

The derivative of x³ is 3x².

The derivative of -4x² is -8x (since -42= -8 and x² becomes x^(2-1)=x).

The derivative of 2x is 2.

The derivative of the constant term -5 is 0.

So, f₂’(x) = 3x² - 8x + 2.

Now, adding f₁’(x) and f₂’(x) together:

f’(x) = f₁’(x) + f₂’(x) = (4x - 3) + (3x² - 8x + 2).

Simplifying this:

Combine like terms:

4x - 8x = -4x

-3 + 2 = -1

So, f’(x) = 3x² - 4x - 1.

Finally, to find f’(3), substitute x = 3 into the derivative:

f’(3) = 3(3)² - 4(3) - 1 = 39 - 12 - 1 = 27 - 12 - 1 = 14.

Therefore, the derivative of f(x) at x₀ = 3 is 14.

Step-by-Step Explanation:

1. Understand the Problem: We are given a function f(x) which is the sum of two polynomials, f₁(x) and f₂(x). We need to find the derivative of f(x) at a specific point, x₀ = 3.

2. Differentiate Each Function Separately: Since differentiation is linear, the derivative of the sum is the sum of the derivatives. So, we can find f₁’(x) and f₂’(x) separately and then add them together.

3. Differentiate f₁(x): f₁(x) = 2x² - 3x + 1.

- The derivative of 2x² is 4x.

- The derivative of -3x is -3.

- The derivative of 1 is 0.

- Therefore, f₁’(x) = 4x - 3.

4. Differentiate f₂(x): f₂(x) = x³ - 4x² + 2x - 5.

- The derivative of x³ is 3x².

- The derivative of -4x² is -8x.

- The derivative of 2x is 2.

- The derivative of -5 is 0.

- Therefore, f₂’(x) = 3x² - 8x + 2.

5. Combine the Derivatives: Now, add f₁’(x) and f₂’(x) together.

-