# The Chain Rule definition and examples

Within the differentiation chapter, the **chain rule** represents a formula of finding the derivative of a composition of two or more functions. The **chain rule derivative** says that: We have 2 functions g and h. We know that g is a function of h and h is a function of x. The derivative of g with respect to x represents the derivative of g(h) with respect to h times the derivative of h(x) with respect to x.

The **derivatives chain rule** may seem very difficult, but in fact it is nothing but a simple mathematical formula. In order to prove how simple and easy to understand is this practice, we will provide you some **chain rule derivatives**. Note about **chain rule integration**: The **integral chain rule** is defined as the U-Substitution which is the counter part of the chain rule.

## Chain rule examples

It is too much important for you to study this rule in brief, this is why we are going to offer a series of **chain rule examples**.

(1) The first example will contain the **multivariable chain rule**. Pay attention so that you will be able to use this rule later.

Problem: Let z = x^{3}y – y^{3} where x and y are parameterized as x = p^{2} and y = 3p. Find dz/dp:

We know that Putting the values of x, y,

(2) The next **chain rule example** refers to **partial derivative chain rule**.

Problem: Let p = e^{xy}, where x(u, v) = uv and y(u, v) = 1/v . Find and :

We know that

Also we have

(3) We saw how the **chain rule partial derivatives** work, now it's time to take a look at the **antiderivative chain rule**. Practically, in this situation we will use the **reverse chain rule**.

Problem: Find the antiderivative of

(4) After all these examples of **derivatives chain rules**, now we will talk about **double chain rule**. This rule is used when you want to find the derivative of a function. In this situation there is a function under the main function. Practically, we have to differentiate all the functions according to their method or according to the sequence.

## Solve some Chain Rule Practice Problems

This chapter in math is actually a little bit difficult, this is why practice is required. Solve the following **chain rule practice problems **and test what you have learn from this lesson to increase the **chain rule probability **of understanding.

Problem 1: Using chain rule, find of y = log (sin x).

Problem 2: Using reverse chain rule, integrate cos(x^{2} + 4)dx.

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