# What is the second derivative of #f(x)=sin(x^2) #?

##### 1 Answer

#### Explanation:

The first derivative of the function can be found using the chain rule. The chain rule states that when differentiating a function that contains another function inside of it, you should differentiate the outside function while keeping the inside function intact and then multiply that by the derivative of the inside function.

To formalize this, this is written as

In the case of a sine function, as we have here, the chain rule applies as follows:

Here, since we are differentiating

#f'(x)=cos(x^2)*d/dx[x^2]=2xcos(x^2)#

To find the second derivative, we will this time have to use the product rule, since we're multiplying two different functions.

The product rule states that

When we apply this to

#f''(x)=cos(x^2)d/dx[2x]+2xd/dx[cos(x^2)]#

Here, we have *almost* did the exact same derivative to find the first derivative of the function. This time, though, we have to deal with a cosine function instead of a sine function.

Since the derivative of

This means that

Plug both of the derivatives back into the equation for

#f''(x)=2cos(x^2)-4x^2sin(x^2)#