Chain rule of differentiation. Today I will talk about the chain rule of differentiation. Here it goes.

### Chain rule of differentiation

Suppose is a function of . And, is a function of .

This means . Again, .

And, my task is to find the value of .

Here I’ll use the chain rule to get the value of .

Thus it will be,

See also: Formulas for differentiation

There are two other rules in differentiation like product rule & quotient rule.

Also, I have already written on the differentiation of different kinds of functions like implicit functions, logarithmic functions and parametric functions.

Now I’ll give you some examples of the chain rule.

#### Some examples of chain rule of differentiation

Here are some examples of the chain rule.

Disclaimer: None of these examples is mine. I have chosen these from some book or books. The references are at the end of the post.

##### Example 1

According to Croft et al. (2000) “Use the chain rule to differentiate the following: .”

##### Solution

Here I have to differentiate the function .

So, I’ll give it a name first. Let’s say .

Also, I’ll bring a new function, say, . Now the function is

Therefore the function will be .

So, now I have two functions. One is

(1)

(2)

Thus I’ll use the chain rule of differentiation to get the value of .

First of all, I’ll differentiate equation (1) with respect to .

Thus I get

This gives

Now I’ll differentiate equation (2) with respect to .

Thus I get

This gives

So the value of will be

Thus I get

Now I’ll replacec with to get

Next I’ll simplify it a bit to get

Hence I can conclude that is the answer to this example.

Now I’ll go to the next example.

##### Example 2

According to Croft et al. (2000) “Differentiate … the following function: .”

##### Solution

Here I have to differentiate the function .

Also, I’ll bring a new function, say, . Now the function is

Therefore the function will be .

So, now I have two functions. One is

(3)

(4)

Thus I’ll use the chain rule to get the value of .

First of all, I’ll differentiate equation (3) with respect to .

Thus I get

This gives

Now I’ll differentiate equation (4) with respect to .

Thus I get

This gives

So the value of will be

Thus I get

Now I’ll replacec with to get

Hence I can conclude that is the answer to this example.

Now I’ll go to the next example.

##### Example 3

According to Croft et al. (2000) “Differentiate … the following function: .”

##### Solution

Here I have to differentiate the function .

Also, I’ll bring a new function, say, . Now the function is

Therefore the function will be .

So, now I have two functions. One is

(5)

(6)

Thus I’ll use the chain rule to get the value of .

First of all, I’ll differentiate equation (5) with respect to .

Thus I get

This gives

Now I’ll differentiate equation (6) with respect to .

Thus I get

This gives

So the value of will be

Thus I get

Now I’ll replacec with to get

Hence I can conclude that is the answer to this example.

Dear friends, this is the end of today’s post. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

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