# If f(x) + x^{2} [f(x)]^{4} = 18 and f(1) = 2, find f'(1).

**Solution.**

Differentiating the given equation w.r.t x we get:

f’(x) + (2x)[f(x)]^{4} + x^{2}(4)[f(x)]³f’(x) = 0

Substituting the value of x as 1 in the above equation we get

f’(1) + (2(1))[f(1)]⁴ + (1)^{2}(4)[f(1)]^{3}f'(1) = 0

f(1) = 2

f’(1) + 2 [2]^{4} + 4(2)^{3}f’(1) = 0

f’(1) + 32 + 4(8)f’(1) = 0

33f’(1) = -32

**f’(1) = -32/33**

**Hence, the required value is -32/33.**

## If f(x) + x^{2} [f(x)]^{4} = 18 and f(1) = 2, find f'(1).

**Summary:**

Differentiating and solving the resulting equation the value of f’(1) = -32/33.