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

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The value of f'(1) has to be determined given that f(x) + x^(2)(f(x))^3 = 10 and f(1) = 2.

f(x) + x^(2)(f(x))^3= 10

Differentiating with respect to x gives

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

f'(x) = -(1 + 6x*(f(x))^2)/(x^2*(f(x))^3)

f'(1) = -(1 + 6*4)/(1*8)

=> -11/8

The value of f'(1) = -11/8

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