Question

hi! trying to find a difference quotient, but how do I approach this when there are fractions in the numerator of a bigger fraction?

Answer 1

$DQ=\frac{\frac{1}{2x+2h}-\frac{1}{2x}}{h}$

start by solving the subtraction of fractions in the numerator using the subtraction of fractions

$\frac{a}{b}-\frac{c}{d}=\frac{a\cdot d-b\cdot c}{bd}$

Then, apply to the fractions in the numerator

$\frac{1}{2x+2h}-\frac{1}{2x}=\frac{2x-(2x+2h)}{2x\cdot(2x+2h)}$

Simplify the expression at the numerator

$\frac{2h}{4x^2+4xh}$

$DQ=\frac{\frac{2x-2x-2h}{4x^2+4hx}}{h}$

simplify

$DQ=\frac{-\frac{h}{2x^2+2xh}}{h}$

then apply the division of fractions

$\begin{gathered} DQ=\frac{-h}{h\cdot(2x^2+2xh)} \\ DQ=-\frac{1}{2x^2+2xh} \end{gathered}$

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