What is the second derivative of #f(x)=sin(x^2) #?
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
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
Here, we have
Since the derivative of
This means that
Plug both of the derivatives back into the equation for