Derivation problems that involve the composition of functions can be solved using the chain rule formula. This formula allows us to derive a composition of functions such as but not limited to *f(g(x))*.

Here, we will look at a summary of the chain rule. Additionally, we will explore several examples with answers to understand the application of the chain rule formula.

## Summary of The Chain Rule

The chain rule is a very helpful tool used to derive a composition of different functions. It is a rule that states that the derivative of a composition of at least two different types of functions is equal to the derivative of the outer function *f(u) *multiplied by the derivative of the inner function *g(x),* where* u = g(x) *as a domain of *f(u)*.

This gives us the chain rule formula as:

or in a another form, it can be illustrated as:

where

- the outer function
- , the domain of outer function
- the derivative of the outer function in terms of
- the derivative of inner function in terms of

We use this formula to derive functions that have the following forms:

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## Chain Rule – Examples with Answers

Using the formula detailed above, we can derive various functions that are written as compositions. Each of the following examples has its respective detailed solution. It is recommended for you to try to solve the sample problems yourself before looking at the solution.

**EXAMPLE 1**

Find the derivative of .

##### Solution

The first thing we need to do is to list down the chain rule formula for our reference:

If you are a beginner, it is recommended that you identify the functions involved in the composition. Otherwise, you can directly use the chain rule formula with fewer steps as long as you know the derivative methods of the type of functions involved.

Assuming you are a beginner, let us identify the functions involved from the function composition:

The given is

From the given, we have

If , then

∴ is a power function and will use the power formula to be derived

∴ is an polynomial function and will use the sum/difference of derivatives to be derived

If , then

Applying the chain rule formula, we have:

Since , let’s substitute into :

Simplifying algebraically, we have

**And the final answer is:**

**EXAMPLE 2**

Find the derivative of .

##### Solution

Let us first list down the chain rule formula for our reference:

Let’s identify the functions involved from the function composition:

The given is

Since this is a radical function, it is always recommended to rewrite it from radical to exponent form to be derivable. Rewriting, we have

From the given, we have

If , then

∴ is a power function and will use the power formula to be derived

∴ is an polynomial function and will use the sum/difference of derivatives to be derived

If , then

Applying the chain rule formula, we have:

Since , let’s substitute into :

Simplifying algebraically, we have

**And the final answer is:**

*in radical form*

**EXAMPLE 3**

Find the derivative of .

##### Solution

If , then

∴ is a trigonometric function and will use the derivative of trigonometric functions to be derived

∴ is an polynomial function and will use the sum/difference of derivatives to be derived

If , then

Applying the chain rule formula, we have:

Since , let’s substitute into :

Simplifying algebraically, we have

**And the final answer is:**

**EXAMPLE 4**

Find the the derivative of

##### Solution

If , then

∴ is a power function and will use the power formula to be derived

∴ is a trigonometric function and will use the derivative of trigonometric functions to be derived

If , then

Applying the chain rule formula, we have:

Since , let’s substitute into :

Simplifying algebraically, we have

**And the final answer is:**

**EXAMPLE 5**

Find the derivative of .

##### Solution

If , then

∴ is a logarithmic function and will use the derivative of logarithmic functions to be derived

∴ is a sum of power and exponential functions and will use the sum/difference of derivatives to be derived

If , then

Applying the chain rule formula, we have:

Since , let’s substitute into :

Simplifying algebraically, we have

**And the final answer is:**

**EXAMPLE 6**

Find the derivative of .

##### Solution

If , then

∴ is an inverse trigonometric function and will use the derivative of inverse trigonometric functions to be derived

∴ is a rational function and will use the quotient rule to be derived

If , then

Applying the chain rule formula, we have:

Since , let’s substitute into :

Simplifying algebraically, we have

**And the final answer is:**

**EXAMPLE 7**

Find the derivative of

##### Solution

This is a *more complex case* as the function is a *composition of four functions*. From the given, we have

If , then

∴ is a power function and will use power formula to be derived

If , then

∴ is a trigonometric function and will use the derivative of trigonometric functions to be derived

If , then

∴ is a exponential function and will use the derivative of exponential functions to be derived

∴ is a monomial function and will use power formula to be derived

If , then

If , then

If , then

Adjusting our *chain rule formula* for the derivative of *compositions of four functions*, we have

Applying our adjusted chain rule formula for the derivative of composition of four functions, we have

Since , and , let’s do the substitutions:

Simplifying algebraically, we have

**And the final answer is:**

As you can observe from our solution in this problem, deriving compositions of four functions, you will realize why the chain rule is coined from the term “chain”.

## Chain Rule – Practice problems

Solve the following derivation problems and test your knowledge on this topic. Use the chain rule formula detailed above to solve the exercises. If you have problems with these exercises, you can study the examples solved above.

## See also

Interested in learning more about the chain rule? Take a look at these pages:

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