Problem 40 - College Algebra and Calculus: An Applied Approach

Question

Use the General Power Rule to find the derivative of the function. $$ f(x)=(4-3 x)^{-5 / 2} $$

Step-by-Step Solution

Answer

So, the derivative of \((4-3x)^{-5/2}\) is \(15/2 \cdot (4-3x)^{-7/2}\)

Step 1: Identify the Internal and External Functions

Split the function into an inside function and an outside function. Here, the inside function (also called g(x)) is \(4 - 3x\) and the outside function, denoted as f(g(x)), is \((g)^{-5/2}\).

Step 2: Apply the Chain Rule to find the Derivative

The Chain Rule states that the derivative of a composite function is the derivative of the outside function evaluated at the inside function, times the derivative of the inside function by itself. Express it as: \((f(g(x)))' = f'(g(x)) \cdot g'(x)\).

Step 3: Apply the Power Rule to the Outside Function

The Power Rule states that the derivative of \(x^n\) is \(nx^{n-1}\). Applying this rule to the outside function \((g)^{-5/2}\), you find its derivative as \(f'(g) = -5/2 \cdot g^{-5/2 - 1} = -5/2 \cdot g^{-7/2}\). Now replace \(g\) with the inside function: \(f'(g) = -5/2 \cdot (4-3x)^{-7/2}\)

Step 4: Find the Derivative of the Inside Function

The inside function is \(4 - 3x\). Its derivative is fairly simple, the derivative of a constant is 0, and the derivative of \(-3x\) is \(-3\). So \(g'(x) = -3\)

Step 5: Assemble the Full Derivative

Put all pieces together. The full derivative of the original function using the Chain Rule is \(f'(x) = f'(g(x)) \cdot g'(x) = -5/2 \cdot (4-3x)^{-7/2} \cdot -3 = 15/2 \cdot (4-3x)^{-7/2} \).