If 30+𝑓(𝑥)+𝑥2(𝑓(𝑥))3=0 and 𝑓(−1)=−3, find 𝑓′(−1).

Question

If 30+𝑓(𝑥)+𝑥2(𝑓(𝑥))3=0 and 𝑓(−1)=−3, find 𝑓′(−1).

Answer 1

30 + 𝑓(𝑥) + 𝑥²(𝑓(𝑥))³ = 0

Take the derivative using the chain rule and the product rule:

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

at x = -1:

f'(-1) + (-1)²(3(f(-1))²)(f'(-1)) + 2(-1)(f(-1))³ = 0

Substitue 𝑓(−1) = −3 and simplify:

f'(-1) + (-1)²(3(-3)²(f'(-1)) + 2(-1)(-3)³ = 0

f'(-1) + 27(f'(-1)) + 54 = 0

28 (f'(-1)) = -54

𝑓′(−1) = -54 / 28 = -27 / 14 = -1.9286