The Given Pair of Functions

Oct 13, 2025 by windailys.com 82 views

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Introduction

In calculus, the chain rule is a fundamental concept used to find the derivative of a composite function. It is a powerful tool that allows us to differentiate functions of functions, which is essential in various fields such as physics, engineering, and economics. In this article, we will explore how to find the derivative of a composite function using the chain rule.

The Chain Rule

The chain rule is a formula for finding the derivative of a composite function. It states that if we have a function of the form $y = f(u)$ , where $u = g(x)$ , then the derivative of $y$ with respect to $x$ is given by:

\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}

This formula can be interpreted as follows: the derivative of $y$ with respect to $x$ is equal to the derivative of $y$ with respect to $u$ multiplied by the derivative of $u$ with respect to $x$ .

The Given Pair of Functions

We are given the pair of functions:

y = 5u^2 + 3u

u = x^3 + 1

Our goal is to find the derivative of $y$ with respect to $x$ using the chain rule.

Finding the Derivative of $y$ with Respect to $u$

To find the derivative of $y$ with respect to $u$ , we will differentiate the function $y = 5u^2 + 3u$ with respect to $u$ . Using the power rule, we get:

\frac{dy}{du} = 10u + 3

Finding the Derivative of $u$ with Respect to $x$

To find the derivative of $u$ with respect to $x$ , we will differentiate the function $u = x^3 + 1$ with respect to $x$ . Using the power rule, we get:

\frac{du}{dx} = 3x^2

Applying the Chain Rule

Now that we have found the derivatives of $y$ with respect to $u$ and $u$ with respect to $x$ , we can apply the chain rule to find the derivative of $y$ with respect to $x$ :

\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}

\frac{dy}{dx} = (10u + 3) \cdot 3x^2

Substituting the Value of $u$

We are given that $u = x^3 + 1$ . We can substitute this value into the expression for $\frac{dy}{dx}$ :

\frac{dy}{dx} = (10(x^3 + 1) + 3) \cdot 3x^2

Simplifying the Expression

We can simplify the expression for $\frac{dy}{dx}$ by distributing the $3x^2$ term:

\frac{dy}{dx} = 30x^3 + 30 + 9x^2

Conclusion

In this article, we have used the chain rule to find the derivative of a composite function. We have shown that the derivative of $y$ with respect to $x$ is given by:

\frac{dy}{dx} = 30x^3 + 30 + 9x^2

This result can be used to solve a variety of problems in calculus and other fields.

Example Problems

Problem 1

Find the derivative of the function $y = 2u^3 + u^2$ , where $u = x^2 + 1$ .

Solution

To find the derivative of $y$ with respect to $x$ , we will first find the derivative of $y$ with respect to $u$ :

\frac{dy}{du} = 6u^2 + 2u

Next, we will find the derivative of $u$ with respect to $x$ :

\frac{du}{dx} = 2x

Finally, we will apply the chain rule to find the derivative of $y$ with respect to $x$ :

\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}

\frac{dy}{dx} = (6u^2 + 2u) \cdot 2x

Substituting the value of $u$ , we get:

\frac{dy}{dx} = (6(x^2 + 1)^2 + 2(x^2 + 1)) \cdot 2x

Simplifying the expression, we get:

\frac{dy}{dx} = 12x^4 + 24x^2 + 12x + 4x^2 + 2x

\frac{dy}{dx} = 12x^4 + 26x^2 + 14x

Problem 2

Find the derivative of the function $y = 3u^2 + 2u$ , where $u = x^4 + 1$ .