Problem 55 - College Algebra and Calculus: An Applied Approach

Question

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ f(x)=\frac{1}{\left(x^{2}-3 x\right)^{2}} $$

Step-by-Step Solution

Answer

The derivative of the given function is \(f'(x)= \frac{2*(2x-3)}{(x^{2}-3x)^{3}}\)

Step 1: Identify the Inner and Outer Functions

A key step in solving this problem is determining the inner and outer functions. Here, the outer function is \(1/u^{2}\) and the inner function is \(u=x^{2}-3x\).

Step 2: Apply Chain Rule

Apply the Chain Rule, which states that the derivative of a composite function is the derivative of the outer function, times the derivative of the inner function. Here, the derivative of the outer function is \(-2/u^{3}\) and the derivative of the inner function (found by applying the Power Rule) is \(2x-3\).

Step 3: Substitute the Derivatives into Chain Rule

Substitute the derived expressions into the Chain Rule equation. We have \((-2/u^{3})*(2x-3)\). Now, substitute \(u=x^{2}-3x\) back into the equation to get the final answer.

Step 4: Simplify the Result

Simplify the results to get the final expression. The derivative \[f'(x)= \frac{2*(2x-3)}{(x^{2}-3x)^{3}}\]