uri2e4g

2022-07-09

Derivatives of a fraction function
An example of a fraction function is:
$y=\frac{-8x}{\left({x}^{2}+3{\right)}^{2}}$
The quotient rule says that if the function one wishes to differentiate, $f\left(x\right)$, can be written as:
$h\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$
Then the derivative is (according to what I learned):
${h}^{\prime }\left(x\right)=\frac{{f}^{\prime }\left(x\right)g\left(x\right)-f\left(x\right){g}^{\prime }\left(x\right)}{\left(g\left(x\right){\right)}^{2}}$
Then I think the procedure is the following:
$\begin{array}{rl}{y}^{\prime }& =\frac{24\left({x}^{2}-1\right)}{\left(\left({x}^{2}+3{\right)}^{2}{\right)}^{2}}\\ & =\frac{24\left({x}^{2}-1\right)}{\left({x}^{2}+3{\right)}^{4}}\end{array}$
However, the solution is...
${y}^{\prime }=\frac{24\left({x}^{2}-1\right)}{\left({x}^{2}+3{\right)}^{3}}$
What are my mistakes?
What is the correct way to derivate fractions?

Jayvion Tyler

Expert

Your second step (after writing down the quotient rule) should be:
${y}^{\prime }=\frac{-8\left({x}^{2}+3{\right)}^{2}+8x\cdot 2\cdot 2x\left({x}^{2}+3\right)}{\left(\left({x}^{2}+3{\right)}^{2}{\right)}^{2}},$
and then an ${x}^{2}+3$ cancels off and gives you the correct answer:
$\frac{-8\left({x}^{2}+3{\right)}^{2}+8x\cdot 2\cdot 2x\left({x}^{2}+3\right)}{\left(\left({x}^{2}+3{\right)}^{2}{\right)}^{2}}=\frac{-8\left({x}^{2}+3\right)+32{x}^{2}}{\left({x}^{2}+3{\right)}^{3}}=\frac{24\left({x}^{2}-1\right)}{\left({x}^{2}+3{\right)}^{3}}.$

Rebecca Villa

Expert

Hint #1: If you write your fraction as
$HIGH/LOW,$
then the derivative for the quotient is "given" by the mnemonic
$\frac{LOW\cdot d\left(HIGH\right)-HIGH\cdot d\left(LOW\right)}{\left(LOW{\right)}^{2}}.$
Hint #2: $d\left(HIGH\right)=-8$
Hint #3: $d\left(LOW\right)=2\left({x}^{2}+3\right)\left(2x\right)=4x\left({x}^{2}+3\right)$
Can you take it from here?

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