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

To find f '(1), we first need to differentiate the given equation with respect to x.

The given equation is f(x) + x^2[f(x)]^3 = 10.

Differentiating both sides with respect to x, we get:

f'(x) + 2x[f(x)]^3 + 3x^2[f(x)]^2*f'(x) = 0.

We know that f(1) = 2, so we substitute x = 1 and f(1) = 2 into the differentiated equation:

f'(1) + 22^3 + 312^2f'(1) = 0.

Solving this equation for f'(1), we get:

f'(1) = -16/(1 + 12) = -16/13.

So, f '(1) = -16/13.

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