If f(x) + x² [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 + x²(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)]³f'(1) = 0

f(1) = 2

f’(1) + 2 [2]⁴ + 4(2)³f’(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) + x² [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.