Question

If f(x) + x^2[f(x)]^3 = 10 and f(1) = 2. What is f'(1)

Answer 1

(x) + x²[f(x)]³ = 10

f(1) = 2     (when x=1 , f(x)=2)

2 + 1²(2)³ = 10

2 + 8 = 10

10 = 10

Evaluate the derivative of f(x) using the point (1, 2)

Let f(x) = y

y + x²y³ = 10

Derive both sides

y' + 2xy³ + 3x²y²y' = 0

y' + 3x²y²y' = -2xy³

y' (1 + 3x²y²) = -2xy³

f'(x) = (-2xy³) / (1 + 3x²y²)

So what is  f'(x) at x=1.

f'(1) = (-2(2³)) / (1 + 3(2²))

f'(1) = -16 / (1 + 12)

f'(1) = -16 / 13