Given integral, find derivative. Help please!

Question

If f(x) = ∫ [from 3 to x⁷+8x⁴+6] cos(t)dt

then f'(x) = ?

Could someone please help me understand the steps required to solve this? Thanks so much!

Answer 1

If c is a constant and if f(x) = ∫_{(c to g(x))} h(t)dt, then by the Fundamental Theorem of Calculus,

f'(x) = h(g(x))g'(x)

So, for f(x) = ∫_{(3 to x^7+8x^4+6)} cos t dt = cos(x⁷+8x⁴+6)(7x⁶+32x³)

Answer 2

f'(x) = cos(x⁷ + 8x⁴ + 6)·(7x⁶ + 32x³) by 2nd fundamental theorem of calculus