Question

If f(x) = (x²-3)⁴, then f'(1) =___

Answer 1

To calculate f'(1) for the function f(x) = (x²-3)⁴, apply the chain rule of differentiation, leading to f'(1) = -64 after evaluating the derivative at x=1.

Step-by-step explanation:

To find f'(1) for the function f(x) = (x²-3)⁴, we need to use the chain rule of differentiation. This rule states that if a function u(x) is raised to a power, its derivative is the power multiplied by the derivative of u(x), all multiplied by u(x) to the power of one less than the original.

Let's apply the chain rule:

1. Compute the inside function's derivative: u'(x) = 2x where u(x) = x² - 3.
2. Multiply this derivative by the derivative of the outer function with respect to u, which is 4(u³).
3. Evaluate this at x = 1: f'(1) = 4((1²-3)³)(2*1).
4. Calculate the numerical value: f'(1) = 4((-2)³)(2) = -64.

Therefore, f'(1) = -64.