Difference Quotient - Mathwords

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Another Example

This example uses a rational function instead of a polynomial, requiring fraction arithmetic to simplify — a common variation students encounter.

Problem: Find and simplify the difference quotient for f(x) = 1/x.Step 1: Compute f(x + h).f(x+h)=1x+hf(x+h) = \frac{1}{x+h}Step 2: Set up the difference quotient and substitute.f(x+h)−f(x)h=1x+h−1xh\frac{f(x+h) - f(x)}{h} = \frac{\frac{1}{x+h} - \frac{1}{x}}{h}Step 3: Combine the two fractions in the numerator using a common denominator of x(x + h).x−(x+h)x(x+h)h=−hx(x+h)h\frac{\frac{x - (x+h)}{x(x+h)}}{h} = \frac{\frac{-h}{x(x+h)}}{h}Step 4: Dividing by h is the same as multiplying by 1/h. The h's cancel.−hx(x+h)⋅1h=−1x(x+h)\frac{-h}{x(x+h)} \cdot \frac{1}{h} = \frac{-1}{x(x+h)}Answer: The simplified difference quotient is −1 / [x(x + h)].

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