Problem 25 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

The derivative of the function \(g(x)=(4-2x)^3\) is \(-6(4-2x)^2\).

Step 1: Identify the outer and inner functions

Here, the inner function \(g(x)\) is \(4-2x\), and the outer function \(f(u)\) is \(u^3\).

Step 2: Compute the derivative of the outer function

Using the power rule, the derivative of \(u^3\) is \(3u^2\).

Step 3: Compute the derivative of the inner function

The derivative of \(4-2x\) is \(-2\).

Step 4: Apply the chain rule

The chain rule says that the derivative of the composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. So, the derivative is \(3(4-2x)^2\) times \(-2\).

Step 5: Simplify the expression

Multiplying \(3(4-2x)^2\) by \(-2\), we get \(-6(4-2x)^2\). This is the derivative of function \(g(x)=(4-2x)^3\).