Problem 75 - Precalculus Essentials

Question

Find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$for the given function. $$f(x)=x^{2}$$

Step-by-Step Solution

Answer

The simplified difference quotient of the function \(f(x)=x^{2}\) is \(2x+h\).

Step 1: Substitute in the Function

First, substitute \(f(x + h)\) and \(f(x)\) into the difference quotient. This gives \(\frac{(x+h)^{2}-x^{2}}{h}\).

Step 2: Expand and Simplify

Next, simplify the numerator. Expand \((x+h)^{2}\) to \(x^{2}+2hx+h^2\). This gives \(\frac{x^{2}+2hx+h^2-x^{2}}{h}\). The \(x^{2}\) terms will cancel, leaving \(\frac{2hx+h^2}{h}\).

Step 3: Cancel Common Factor

Finally, simplify \(\frac{2hx+h^2}{h}\) by dividing the numerator by the common factor \(h\). This leaves us with \(2x+h\).