If 34+9f(x)+x^2(f(x))3=0 and f(−5)=−1, find f′(−5).

Answer 1

Implicit derivative problem

34+9f(x)+x²f³(x)=0

Taking the derivative

9f '(x) + x²•(3f²(x))f '(x) +2xf³(x) = 0

Solve for f '(x):

f '(x)(9+3x²f²(x)) +2xf³(x)=0

f '(x) = -2xf³(x)/(9+3x²f²(x))

f '(-5) = -2(-5)(-1)³/(9+3(-5)²(-1)²)

= -10/(9+75)

= -10/84