Problem 91 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The simplified difference quotient for the function \(f(x) = \sqrt{x}\) is \(\frac{1}{\sqrt{x+h}+\sqrt{x}}\).

Step 1: Substitution

Substitute \(x + h\) and \(x\) into \(f(x)=\sqrt{x}\) to get \(f(x+h)=\sqrt{x+h}\) and \(f(x)=\sqrt{x}\).

Step 2: Set Up the Difference Quotient

Replace \(f(x+h)\) and \(f(x)\) in the difference quotient \(\frac{f(x+h)-f(x)}{h}\) to get \(\frac{\sqrt{x+h}-\sqrt{x}}{h}\).

Step 3: Rationalize the Denominator

Multiply the expression by \(\frac{\sqrt{x+h}+\sqrt{x}}{\sqrt{x+h}+\sqrt{x}}\) to rationalize the denominator. This results to \(\frac{(x+h)-x}{h(\sqrt{x+h}+\sqrt{x})}\) which simplifies to \(\frac{h}{h(\sqrt{x+h}+\sqrt{x})}\).

Step 4: Simplify the Expression Further

Simplify the expression further by cancelling out the \(h\) in the numerator and the denominator. This leaves \(\frac{1}{\sqrt{x+h}+\sqrt{x}}\).