If f(x) + x² [f(x)]⁴ = 18 and f(1) = 2, find f'(1)

Functions form the backbone of many topics like calculus and algebra. Let's solve a problem related to functions.

Answer: If f(x) + x² [f(x)]⁴ = 18 and f(1) = 2, f'(1) = -32 / 33.

Let's understand the solution in detail.

Explanation:

Note: The derivative of f(x) is represented as f'(x).

To find f'(1) in the given problem, we first differentiate the equation on both sides.

We use the chain rule and multiplication rule of derivatives to differentiate the equation f(x) + x² [f(x)]⁴ = 18

⇒ f'(x) + 2x [f(x)]⁴ + 4x² [f(x)]³ f'(x) = 0

Rearranging the above equation we get:

⇒ f'(x) = -2x [f(x)]⁴ / (1 + 4x² [f(x)]³)

Now, we substitute x = 1 in the above equation.

⇒ f'(1) = -2(1)[f(1)]⁴ / (1 + 4(1)² [f(1)]³)

= -2⁵ / 33 = -32 / 33

Hence, If f(x) + x²[f(x)]⁴ = 18 and f(1) = 2, f'(1) = -32 / 33.