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Chapter 7: Problem 28

Use the General Power Rule to find the derivative of the function. $$ f(x)=\left(4 x-x^{2}\right)^{3} $$

Short Answer

Expert verified
The derivative of the function \(f(x) = (4x - x^2)^3\) using the General Power Rule is \(f'(x) = 12 * [(4x - x^2)^2 * (4 - 2x)]\).

Step by step solution

01

Identify the General Power Rule and Chain Rule

The General Power Rule applied in this context is: if \(f(x) = [u(x)]^n\), then \(f'(x) = n * u^(n - 1) * u'(x)\). Here, \(u(x) = 4x - x^2\) and \(n = 3\). The derivative of the function \(u(x)\) will be obtained using the Power Rule for derivatives, \(f'(x) = n*x^(n-1)\)
02

Compute the derivative of \(u(x)\)

We find the derivative of \(u(x) = 4x - x^2\) as \(u'(x) = 4 - 2x\)
03

Substitute \(u(x)\) and \(u'(x)\) into the General Power Rule

Substitute the computed values into the formula, \(f'(x) = 3 * (4x - x^2)^(3 - 1) * (4 - 2x)\), which becomes \(f'(x) = 3 * (4x - x^2)^2 * (4 - 2x)\)
04

Simplify the derivative

Finally, expand and simplify the expression to get the derivative of \(f(x)\). This leaves us with the final derivative \(f'(x) = 12 * [(4x - x^2)^2 * (4 - 2x)]\)

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