how do you find the derivitive of (sin x^3) and (sin^3 x)?

Question

(sin x^3)= Sin X cubed
(sin^3 x)= Sin cubed X

thanks!

Answer 1

These use the Differentiation Chain Rule

d/dx [sin(x^3) ] = cos(x^3) * d/dx [x^3] = cos(x^3) * (3*(x^2))

d/dx [ (sinx)^3 ] = 3*(sinx)^2 * d/dx [sinx] = 3*(sinx)^2 * cosx

Answer 2

The chain rule is your friend.

You're being asked to differentiate both f(g(x)) and g(f(x)), where f = sinx and g = x^3.

The derivative of f(g(x)) is f'(g(x))*g'(x). The derivative of g(f(x)) = g'(f(x)) * f' (x).

Here f' = cos and g' = 3x^2.

So one of your answers is cos(x^3)*3x^2, while the other is 3sin^2*cos, and you should be careful to figure out which is which.