If f(x) + x2 [f(x)]3 = 10 and f(1) = 2. Find f'(1).

We will be using the concept of differentiation to solve this.

Answer: If f(x) + x2 [f(x)]3 = 10 and f(1) = 2, then f'(1) = -16/13

Let's solve this step by step.

Explanation:

Given that, f(x) + x2 [f(x)]3 = 10 and f(1) = 2

f(x) + x2[f(x)]3 = 10

Differentiate with respect to x on both sides,

f'(x) + {2x [f(x)]3 + 3x2 [f(x)]2 f'(x)} = 0

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

f'(x) {1 + 3x2 [f(x)]2} = -2x [f(x)]3

f'(x)  = -2x [f(x)]3 / {1 + 3x2 [f(x)]2}

Substitue x = 1 and f(1) = 2 above,

f'(1) = -2 (1) [f(1)]3 / {1 + 3 (1)2 [f(1)]2}

f'(1) = -2 [2]3 / {1 + 3 [2]2}

f'(1) = -2[8] / {1 + 3[4]}

f'(1) = -16 / 13

Hence, if f(x) + x2 [f(x)]3 = 10 and f(1) = 2, then f'(1)  = -16/13

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