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# If f(x) + x^{2 }[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) + x^{2 }[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) + x^{2 }[f(x)]^{3} = 10 and f(1) = 2

f(x) + x^{2}[f(x)]^{3} = 10

Differentiate with respect to x on both sides,

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

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

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

f'(x) = -2x [f(x)]^{3} / {1 + 3x^{2 }[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) + x^{2 }[f(x)]^{3} = 10 and f(1) = 2, then f'(1) = -16/13

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