Given f(x) below, what is f′(x)?

Question

Answer 1

The answer to this lies in using fundamental theorem of integration and chain rule of differentiation.

Let F ' (t) = (t^4 -5)^7

Using fundamental theorem of integration:

f(x) = F(x²) -F(2x)

f'(x) = (d/dx)[F(x⁴) -F(2x)]

= (dx⁴/dx)(dF(x⁴)/dx⁴) - (2x)'F'(2x) [ using chain rule of differentiation]

=4x³.((x⁴)⁴ - 5)⁷ -2 ((2x)⁴ -5)⁷

= 4x³.(x¹⁶ - 5)⁷ -2 (16x⁴ -5)⁷ Ans

Answer 2

the integral of f'(x) = f(x)

the derivative of the integral of f'(x) = f(x)

that seems to mean f'(x) = (t^4 -5)^7

but that leaves open the question of what to do with the limits of integration. Someone else may figure that out

f(x) is a definite integral, meaning no need to figure out the constant term when you integrate. My guess is ignore the limits, as you're dealing only with the derivative. This might be a "trick question"