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 = x3y – y3 where x and y are parameterized as x = p2 and y = 3p. Find dz/dp:

We know that multivariable chain rule Putting the values of x, y,

 chain rule examples

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

Problem: Let p = exy, where x(u, v) = uv and y(u, v) = 1/v . Find chain rule and chain rule examples :

We know that chain rule example

Also we have example of chain rule

(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  antiderivative chain rule
reverse chain rule

(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(x2 + 4)dx.

Ian Roberts
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