How Do You Differentiate Complex Functions Involving Exponents and Operations?

[DespicableMe]

Homework Statement

If f(x) can be differentiated, find expressions for the derivatives of the following functions.

a) g(x) = f(x⁶)
b) h(x) = [ f(x)]⁶
c) f(x) = x²/ f(x)

The Attempt at a Solution

a)
b) Use the product rule first then multiply that expression by the expression for the chain rule?
c) Use the quotient rule?

My problem is knowing the difference between what they mean when the exponent is inside the bracket with x, when the exponent is outside f(x) and f(x) alone.

 

[Mark44]

Homework Statement

If f(x) can be differentiated, find expressions for the derivatives of the following functions.

a) g(x) = f(x⁶)
b) h(x) = [ f(x)]⁶
c) f(x) = x²/ f(x)

The Attempt at a Solution

a)

Use the chain rule.

b) Use the product rule first then multiply that expression by the expression for the chain rule?

No. Use the chain rule.

c) Use the quotient rule?

Yes.

My problem is knowing the difference between what they mean when the exponent is inside the bracket with x, when the exponent is outside f(x) and f(x) alone.

For f(x⁶) and (f(x))², the difference is the order in which you evaluate things. As an example, let f(x) = 2x + 1

For f(x⁶), you raise x to the 6th power, and then use that value as the input to your function f. Using my example, f(x⁶) = 2(x⁶) + 1 = 2x⁶ + 1.

For [f(x)]⁶, you use x as the input to the function, and then raise the output of the function to the 6th power. Using my example, [f(x)]⁶ = [2x + 1]⁶ = 64x⁶ + a bunch of other terms.