Problem 28 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

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 1: 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, \(f'(x) = n*x^(n-1)\)

Step 2: Compute the derivative of \(u(x)\)

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

Step 3: 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 \(f'(x) = 3 * (4x - x^2)^2 * (4 - 2x)\)

Step 4: Simplify the derivative

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