Review your rational function differentiation skills and use them to solve problems.

How do I differentiate rational functions?

To differentiate rational functions, all you need is to know how to differentiate polynomials and the quotient rule. Let's differentiate start fraction, x, start superscript, 2, end superscript, minus, 2, divided by, x, minus, 1, end fraction for example.
=ddx(x22x1)=(x1)ddx(x22)(x22)ddx(x1)(x1)2Quotient rule=(x1)(2x)(x22)(1)(x1)2=2x22xx2+2(x1)2=x22x+2(x1)2\begin{aligned} &=\dfrac{d}{dx}\left(\dfrac{x^2-2}{x-1}\right) \\\\ &=\dfrac{(x-1)\dfrac{d}{dx}(x^2-2)-(x^2-2)\dfrac{d}{dx}(x-1)}{(x-1)^2}&&\gray{\text{Quotient rule}} \\\\ &=\dfrac{(x-1)(2x)-(x^2-2)(1)}{(x-1)^2} \\\\ &=\dfrac{2x^2-2x-x^2+2}{(x-1)^2} \\\\ &=\dfrac{x^2-2x+2}{(x-1)^2} \end{aligned}
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Practice set 1: Differentiate rational functions

Problem 1.1
g, left parenthesis, x, right parenthesis, equals, start fraction, x, minus, 5, divided by, x, start superscript, 2, end superscript, plus, 1, end fraction
g, prime, left parenthesis, x, right parenthesis, equals

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Practice set 2: Evaluate rational functions derivatives

Problem 2.1
f, left parenthesis, x, right parenthesis, equals, start fraction, 3, x, start superscript, 2, end superscript, plus, 1, divided by, x, plus, 2, end fraction
f, prime, left parenthesis, minus, 3, right parenthesis, equals, question mark
Please choose from one of the following options.

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