Question

derivative of sin (2x²)​

Answer 1

Explanation:

There are two best ways to do this that I see.

Use the chain rule twice

Use the product rule and the chain rule

Starting with my preferred method #1.

First you want to identify the outermost “chain.” f=x^2, g=sin(x^2)

Then, you move inward. h=sin(x), j=(x^2)

You apply the chain rule to this:

h=sin(x) h’=cos(x)

j=x^2 j’=2x

h’(j(x))*j’(x)=2xcos(x^2).

With this, we move out

f=x^2 f’=2x

g=sin(x^2) g’=2xcos(x^2)

f’(g(x))*g’(x)=4xsin(x^2)cos(x^2)

The derivative of sin^2(x^2)=4xsin(x^2)cos(x^2).

2. Use product and chain rule

sin^2(x^2)=sin(x^2)*sin(x^2)

f=sin(x^2) f’=2xcos(x^2)

g=sin(x^2) g’=2xcos(x^2)

f’g+g’f= 2xcos(x^2)sin(x^2)+2xcos(x^2)sin(x^2)=4xsin(x^2)cos(x^2)

The derivative of sin^2(x^2)=4xsin(x^2)cos(x^2)