If 21+𝑓(𝑥)+5𝑥2(𝑓(𝑥))3=0 and 𝑓(2)=−1, find 𝑓′(2).

Answer 1

21 +f(x) + 5x²(f(x))³=0

Use implicit derivation

0 + f'(x) + 5x²(3(f(x)²)f'(x)+10x(f(x))³ = 0

f'(x)(1 + 15x²(f(x))²) = -10x(f(x))³

f '(x) = -10x(f(x))³/(1 + 15x²(f(x))²)

Now just substitute for x=2 and f(2) = -1 to evaluate f '(2)