f'(x) given a eqaution and f(x) and f(-1) and left to find f'(-1)

Question

The question is : Find f'(-1) if 2x(f(x))^3+4x^5 f(x)=-66 and f(-1)=3 .

Find f'(-1) =

Answer 1

Let’s denote f(x) by y and f’(x) by y’.

We are given that 2x(y^3) + 4x^5 y = -66

Differentiating both sides and using the sum rule on the left hand side (LHS) gives us

d/dx( 2x(y^3)) + d/dx( 4x^5 y ) = d/dx(-66)

Using the product rule on the individual terms on the LHS leads to

2y^3 + 2x.3y^2.y’ + 20x^4.y + 4x^5.y’ = 0

Simplifying and rearranging,

y’ ( 6xy^2 + 4x^5 ) + ( 2y^3 + 20x^4y ) = 0

So, y’ = -( 2y^3 + 20x^4y ) / ( 6xy^2 + 4x^5 )

Substituting x = -1, y = 3 gives us y’ = -( 54 + 60)/( -54 -4)= 114/58 = 57/29