Find the derivative of the function.

Question

Answer 1

This is a chain rule exercise.

Might help to rewrite to make the power rule more evident.

y = sin(x^1/3) + (sin(7x))^1/3

y' = cos(x^1/3) (1/3)x^-2/3 + (1/3)(sin(7x))^-2/3(cos(7x))7

= cos(³√x)/(3 ³√(x²))+7cos(7x)/(3 ³√sin²(7x))

Answer 2

sin(cuberoot(x)) can be written as sin(x^1/3) and using the chain rule the derivative is cos(x^1/3)•(1/3x^-2/3)

cuberoot(sin(7x)) can be written as (sin(7x))^1/3 and, again using the chain rule is 1/3(sin(7x))^-2/3•cos(7x)•7

These can both be simplified a bit and put together to form the answer.