Problem 90 - Precalculus Essentials

Question

Find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$for the given function. $$f(x)=\frac{1}{2 x}$$

Step-by-Step Solution

Answer

The simplified difference quotient for the given function is \(\frac{-1}{(x + h)2}\)

Step 1: Compute \(f(x + h)\)

Substitute \(x + h\) into the function \(f(x) = \frac{1}{2x}\) to find \(f(x + h) = \frac{1}{2(x + h)}\)

Step 2: Substitute Into the Difference Quotient

Now, substitute \(f(x)\) and \(f(x + h)\) into the difference quotient equation to obtain : \(\frac{f(x+h)-f(x)}{h} = \frac{\frac{1}{2(x + h)}-\frac{1}{2x}}{h}\)

Step 3: Simplifying the Difference Quotient

The next step is to simplify the difference quotient. Remove the fraction inside the numerator by multiplying everything by \(2x(x + h)\). This will be done as follows: \[2x(x + h)\left[\frac{1}{2(x + h)} - \frac{1}{2x}\right] \div h(x + h)2x = x - (x + h) \div h(x + h)2x = \frac{x2 - x(x + h)}{h(x + h)2x}\]

Step 4: More Simplifying

After further simplification, the difference quotient will be expressed as: \[ \frac{x^2 - x^2 - hx}{h(x + h)2x} = \frac{-hx}{h(x + h)2x}\] Now, cancel the h in the numerator and denominator to finally simplify the difference quotient as: \[ \frac{-x}{(x + h)2x}\]

Step 5: Simplify Further (Final Step)

Lastly, the x in the denominator and numerator cancel out leaving only : \[ \frac{-1}{(x + h)2}\]