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# Chain Rule: Guide with Theorem and Examples

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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 = e3x

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

and

the outer function is f(u) = eu

Applying the Chain Rule, we get:

dy/dx = 3e3x

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.

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#### Moonpreneur

Moonpreneur is an ed-tech company that imparts tech entrepreneurship to children aged 6 to 15. Its flagship offering, the Innovator Program, offers students a holistic learning experience that blends Technical Skills, Power Skills, and Entrepreneurial Skills with streams such as Robotics, Game Development, App Development, Advanced Math, Scratch Coding, and Book Writing & Publishing.
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