find an equation of the function

Question

Find an equation of the function f that passes through (1,-3) and has a derivative given by f'(x)=12x^3(3x^4-2)^3

Answer 1

If f '(x) = 12x³(3x⁴ - 2)³ then f(x) = 1/4(3x⁴ - 2)⁴ + C because taking the derivative (using the power rule and chain rule) will result in the given derivative (Note that any constant "C" will have a derivative of zero so it must be included).

If f(x) = 1/4(3x⁴ - 2)⁴ + C and f(1) = -3 then we can solve for "C":

-3 = 1/4(3(1)⁴ - 2)⁴ + C

-3 = 1/4(1)⁴ + C

C = -3 - 1/4 = -12/4 - 1/4 = -13/4

So f(x) = 1/4(3x⁴ - 2)⁴ - 13/4