Use the chain rule to find the derivative of the following function.𝑓(𝑥)=4(𝑥3−2𝑥2+2𝑥)100𝑓′(𝑥)=

Sure, here is the step by step solution:

The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

1. Identify the outer function and the inner function. In this case, the outer function is u^100 and the inner function is u = 4(x^3 - 2x^2 + 2x).

2. Find the derivative of the outer function. The derivative of u^100 with respect to u is 100u^99.

3. Find the derivative of the inner function. The derivative of 4(x^3 - 2x^2 + 2x) with respect to x is 12x^2 - 8x + 2.

4. Apply the chain rule. The derivative of the original function is the product of the derivative of the outer function and the derivative of the inner function.

So, f'(x) = 100(4x^3 - 8x^2 + 2x)^99 * (12x^2 - 8x + 2).