NEED HELP URGENT!

Question

I can't figure this out please help.

If 78 + 7f(x) + 2x^2(f(x))^3 = 0 and f(2) = -2, find f '(2).

Answer 1

If 78 + 7f(x) + 2x^2(f(x))^3 = 0 then we can use implicit differentiation to work your problem:

The derivative of 78 = 0

The derivative of 7f(x) = 7•f '(x)

The derivative of (2x²) (f(x))³ is found using the product rule, power rule, and the chain rule. Since the product rule is: (u•v)' = u'v + uv' we can let u = 2x² and v = (f(x))³ so by the power rule u' = 4x and by the power rule and the chain rule v' = 3(f(x))²•f '(x). So putting that together, the derivative of (2x²) (f(x))³ is (4x)((f(x))³ + (2x²)(3(f(x))²•f '(x))

The derivative of 0 is 0. So putting these all together we get:

0 + 7•f '(x) + (4x)((f(x))³ + (2x²)(3(f(x))²•f '(x)) = 0

7•f '(x) + (2x²)(3(f(x))²•f '(x)) = -(4x)((f(x))³

f '(x)[7 + (2x²)(3(f(x))²] = -(4x)((f(x))³

f '(x) = -(4x)((f(x))³/[7 + (2x²)(3(f(x))²]

You can now plug in x = 2 and f(2) = -2 to find f '(2)