Chain Rule: Guide with Theorem and Examples

The Chain Rule is a powerful concept in calculus that provides a systematic way to find the derivatives of complex functions. In this blog post, we’ll break down the Chain Rule, introduce its fundamental theorem, and illustrate its application through easy-to-follow examples.

The Chain Rule Theorem:

Starting with the formal statement of the Chain Rule. If we have a composite function: 

y = f(g(x))

f = inner function

g = outer function

The Chain Rule can be expressed as follows:

dy/dx = dy/dg x dg/dx

In simpler terms, the rate of change of the composite function is equal to the product of the rate of change of the outer function with respect to the inner function, and the rate of change of the inner function with respect to the original variable.

Example 1: Basic Chain Rule Application

Let’s consider the function y = (3x² + 2x)⁵ 

Here, the outer function is f(u) = u⁵

and

the inner function is g(x) = 3x² + 2x 

Applying the Chain Rule, we get:

dy/dx = 5(3x² + 2x)⁴ x (6x + 2) 

Recommended Reading: What is the Derivative of sec x?

Example 2: Chain Rule with Trigonometric Functions

Now, let’s look at y = sin(2x). 

The inner function is g(x) = 2x

and

the outer function is f(u) = sin(u). 

Applying the Chain Rule, we get:

dy/dx = cos(2x) x 2 

Example 3: Chain Rule with Exponential Functions

Consider y = e^3x. 

Here, the inner function is g(x) = 3x

and

the outer function is f(u) = e^u. 

Applying the Chain Rule, we get:

dy/dx = 3e^3x

Recommended Reading: What is cos 5pi 6?

Conclusion:

The Chain Rule is a fundamental tool that simplifies the process of finding derivatives for composite functions. By understanding its theorem and practicing with examples, you’ll gain confidence in differentiating complex mathematical expressions. As you dive deeper into calculus, the Chain Rule will continue to be a reliable companion, enabling you to dissect intricate functions with ease.

Moonpreneur understands the needs and demands this rapidly changing technological world is bringing with it for our kids. Our expert-designed Advanced Math course and Math Quiz for grades 3rd, 4th, 5th, and 6th will help your child develop math skills with hands-on lessons, excite them to learn, and help them build real-life applications. 

Register for a free 60-minute Advanced Math Workshop today!