## Introduction

In calculus, the chain rule is a fundamental concept that allows us to find the derivative of composite functions. It is an essential tool in solving complex calculus problems and is often encountered in homework assignments. In this article, we will explore the 3.1 the chain rule homework, which focuses on applying the chain rule to different functions and understanding its implications. Let's dive in!

## Understanding the Chain Rule

Before we delve into the specific homework problems, let's briefly review the chain rule. The chain rule states that if we have a composite function, f(g(x)), where f and g are differentiable functions, then the derivative of f(g(x)) with respect to x is given by:

d(f(g(x)))/dx = f'(g(x)) * g'(x)

This means that to find the derivative of a composite function, we need to take the derivative of the outer function, evaluated at the inner function, multiplied by the derivative of the inner function.

## Homework Problem 1: Differentiating Trigonometric Functions

### Problem Statement

Find the derivative of the function f(x) = sin(3x^2).

### Solution

To solve this problem, we apply the chain rule. Let's break down the function into two parts: the outer function f(x) = sin(x) and the inner function g(x) = 3x^2. The derivative of the outer function, sin(x), is cos(x). The derivative of the inner function, 3x^2, is 6x. Applying the chain rule, we get:

f'(x) = cos(g(x)) * g'(x) = cos(3x^2) * 6x

So, the derivative of f(x) = sin(3x^2) is f'(x) = 6x * cos(3x^2).

## Homework Problem 2: Chain Rule with Exponential Functions

### Problem Statement

Find the derivative of the function f(x) = e^(2x + 1).

### Solution

Here, we have a composite function with an exponential function as the outer function and a linear function as the inner function. Let's break it down into f(x) = e^g(x), where g(x) = 2x + 1. The derivative of the outer function, e^x, is e^x. The derivative of the inner function, 2x + 1, is 2. Applying the chain rule, we get:

f'(x) = e^(g(x)) * g'(x) = e^(2x + 1) * 2

Therefore, the derivative of f(x) = e^(2x + 1) is f'(x) = 2 * e^(2x + 1).

## Homework Problem 3: Chain Rule with Logarithmic Functions

### Problem Statement

Find the derivative of the function f(x) = ln(5x^3 + 2x).

### Solution

In this problem, we have a composite function with a logarithmic function as the outer function and a polynomial function as the inner function. Let's break it down into f(x) = ln(g(x)), where g(x) = 5x^3 + 2x. The derivative of the outer function, ln(x), is 1/x. The derivative of the inner function, 5x^3 + 2x, is 15x^2 + 2. Applying the chain rule, we get:

f'(x) = (1 / g(x)) * g'(x) = (1 / (5x^3 + 2x)) * (15x^2 + 2)

Therefore, the derivative of f(x) = ln(5x^3 + 2x) is f'(x) = (15x^2 + 2) / (5x^3 + 2x).

## Homework Problem 4: Chain Rule with Composite Functions

### Problem Statement

Find the derivative of the function f(x) = sin(3x^2 + 1).

### Solution

In this problem, we have a composite function with a trigonometric function as the outer function and a polynomial function as the inner function. Let's break it down into f(x) = sin(g(x)), where g(x) = 3x^2 + 1. The derivative of the outer function, sin(x), is cos(x). The derivative of the inner function, 3x^2 + 1, is 6x. Applying the chain rule, we get:

f'(x) = cos(g(x)) * g'(x) = cos(3x^2 + 1) * 6x

Therefore, the derivative of f(x) = sin(3x^2 + 1) is f'(x) = 6x * cos(3x^2 + 1).

## Conclusion

The chain rule is a powerful tool in calculus that allows us to find the derivative of composite functions. By breaking down a composite function into its individual parts and applying the chain rule, we can simplify the process of finding derivatives. In this article, we explored the 3.1 the chain rule homework, which involved differentiating trigonometric, exponential, logarithmic, and composite functions. By understanding and practicing these concepts, we can become more proficient in solving calculus problems that require the application of the chain rule.