Quotient Derivative Example

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Jason76

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\(\displaystyle y = (\dfrac{x^{2} + 3}{x^{2} - 3})^{8}\)

\(\displaystyle y' = 8 (u)^{7} du\)

\(\displaystyle y' = 8 (u)^{7} \dfrac{[x^{2} - 3][2x] - [x^{2} + 3][2x]}{(x^{2} - 3)^{2}}\)

\(\displaystyle y' = 8 (\dfrac{x^{2} + 3}{x^{2} - 3})^{7} \dfrac{[x^{2} - 3][2x] - [x^{2} + 3][2x]}{(x^{2} - 3)^{2}}\)

\(\displaystyle y' = 8 (\dfrac{x^{2} + 3}{x^{2} - 3})^{7} \dfrac{2x^{3} - 6x - 2x^{3} + 6x}{(x^{2} - 3)^{2}}\)

\(\displaystyle y' = 8 (\dfrac{x^{2} + 3}{x^{2} - 3})^{7} \dfrac{0}{(x^{2} - 3)^{2}}\)

\(\displaystyle y' = 8 (\dfrac{x^{2} + 3}{x^{2} - 3})^{7} 0 \)

\(\displaystyle y' = 0 \) :confused:
 
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Jason76 said:
\(\displaystyle y = (\dfrac{x^{2} + 3}{x^{2} - 3})^{8}\)

\(\displaystyle y' = 8 (u)^{7} du\)

\(\displaystyle y' = 8 (u)^{7} \dfrac{[x^{2} - 3][2x] - [x^{2} + 3][2x]}{(x^{2} - 3)^{2}}\)

\(\displaystyle y' = 8 (\dfrac{x^{2} + 3}{x^{2} - 3})^{7} \dfrac{[x^{2} - 3][2x] - [x^{2} + 3][2x]}{(x^{2} - 3)^{2}}\)

\(\displaystyle y' = 8 (\dfrac{x^{2} + 3}{x^{2} - 3})^{7} \dfrac{2x^{3} - 6x - 2x^{3} + 6x}{(x^{2} - 3)^{2}} \ \ \ \ \ \) No, you did not distribute correctly. The far right characters in the numerator should be "- 6x."

\(\displaystyle y' = 8 (\dfrac{x^{2} + 3}{x^{2} - 3})^{7} \dfrac{0}{(x^{2} - 3)^{2}}\)

\(\displaystyle y' = 8 (\dfrac{x^{2} + 3}{x^{2} - 3})^{7} 0 \)

\(\displaystyle y' = 0 \) :confused:
Click to expand...
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\(\displaystyle y = (\dfrac{x^{2} + 3}{x^{2} - 3})^{8}\)

\(\displaystyle y' = 8 (u)^{7} du\)

\(\displaystyle y' = 8 (u)^{7} \dfrac{[x^{2} - 3][2x] - [x^{2} + 3][2x]}{(x^{2} - 3)^{2}}\)

\(\displaystyle y' = 8 (\dfrac{x^{2} + 3}{x^{2} - 3})^{7} \dfrac{[x^{2} - 3][2x] - [x^{2} + 3][2x]}{(x^{2} - 3)^{2}}\)

\(\displaystyle y' = 8 (\dfrac{x^{2} + 3}{x^{2} - 3})^{7} \dfrac{2x^{3} - 6x - 2x^{3} - 6x}{(x^{2} - 3)^{2}}\)

\(\displaystyle y' = 8 (\dfrac{x^{2} + 3}{x^{2} - 3})^{7} \dfrac{-12x}{(x^{2} - 3)^{2}}\)
 
Jason76 said:
\(\displaystyle y = (\dfrac{x^{2} + 3}{x^{2} - 3})^{8}\)

\(\displaystyle y' = 8 (u)^{7} du\)

\(\displaystyle y' = 8 (u)^{7} \dfrac{[x^{2} - 3][2x] - [x^{2} + 3][2x]}{(x^{2} - 3)^{2}}\)

\(\displaystyle y' = 8 (\dfrac{x^{2} + 3}{x^{2} - 3})^{7} \dfrac{[x^{2} - 3][2x] - [x^{2} + 3][2x]}{(x^{2} - 3)^{2}}\)

\(\displaystyle \displaystyle y' = 8 \left (\dfrac{x^{2} + 3}{x^{2} - 3}\right )^{7} \dfrac{2x^{3} - 6x - 2x^{3} - 6x}{(x^{2} - 3)^{2}}\)

\(\displaystyle \displaystyle y' = -96 \left (\dfrac{x^{2} + 3}{x^{2} - 3}\right )^{7} \dfrac{x}{(x^{2} - 3)^{2}}\)
Click to expand...

Try this form

or

\(\displaystyle \displaystyle y' = -96 \dfrac{x*(x^{2} + 3)^7}{(x^{2} - 3)^9} \)
 
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