Let g(x) = sin (cos x^3) Find g ' (x):

The choices are
a) -3x^2sinx^3cos(cos x^3)
b) -3x^2sinx^3sin(cos x^3)
c) -3x^2cosx^3sin(cos x^3)
d) 3x^2sin^2(cos x^3)

I'm not exactly sure where I should start.
Should I begin with d/dx of sin? Or do the inside derivative first...and do I have to separate cos and x^3 as well?

This is a multiple choice question and then the way you should attack the problem should be different than if you were asked to find the derivative of sin[cos(x^3)]

What you do is you use the chain rule, accrding to which the derivative of
sin(f(x)) = cos(f(x)) f'(x)

Without calculkating anything, you immediately see that b) c) and d) cannot be right, so a) must be right.

The prefactor -3x^2sin(x^3) is indeed the derivative of the argument of the sin. You only have to notice that it looks correct and then move on to the next question. In a test you can be given many multiple choice question and then the teacher will test if you can spot the correct answer within seconds.

To find the derivative of g(x) = sin(cos(x^3)), you can start by using the chain rule, which states that if you have a function g(f(x)), then the derivative is g'(f(x)) multiplied by f'(x).

1. Begin by differentiating the outer function, sin(u), where u = cos(x^3). The derivative of sin(u) with respect to u is cos(u).

So far, we have: cos(u)

2. Now, let's differentiate the inner function, u = cos(x^3) with respect to x. Since u is a composition of functions, we need to use the chain rule again.

The derivative of cos(u) with respect to u is -sin(u), and the derivative of u with respect to x is d/dx(cos(x^3)) = -sin(x^3) * 3x^2.

So far, we have: cos(u) * (-sin(x^3) * 3x^2)

3. Multiply the derivatives obtained in step 1 and step 2 to obtain the derivative of g(x) = sin(cos(x^3)).

g'(x) = cos(u) * (-sin(x^3) * 3x^2)

Simplifying further, combining the cos(u) and sin(x^3):
g'(x) = -3x^2 * sin(x^3) * cos(u)

Now, we just need to substitute the value of u back into the expression:

u = cos(x^3)

So, the final answer for g'(x) is:

g'(x) = -3x^2 * sin(x^3) * cos(cos(x^3))

Therefore, the correct choice is c) -3x^2 * cos(x^3) * sin(cos(x^3)).