Using chain rule and quotient rule together

 
 
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Chain rule with quotient rule

Chain rule is also often used with quotient rule.

Let’s look at an example of how these two derivative rules would be used together.

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Applying chain rule as part of quotient rule problems


 
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Quotient rule and chain rule for rational functions

Example

Use chain rule to find the derivative.

???y=\left(\frac{6x^4}{8\sin{x}}\right)^{8}???

If we use substitution, then

???u=\frac{6x^4}{8\sin{x}}???

and according to quotient rule,

???u'=\frac{\left(24x^3\right)(8\sin{x})-\left(6x^4\right)(8\cos{x})}{(8\sin{x})^2}???

Substituting into our original equation, we get

???y=(u)^{8}???

and using power rule with chain rule, the derivative is

???y'=8(u)^{7}(u')???

Back-substituting for ???u??? and ???u'???, the derivative is

???y'=8\left(\frac{6x^4}{8\sin{x}}\right)^7\left[\frac{\left(24x^3\right)(8\sin{x})-\left(6x^4\right)(8\cos{x})}{(8\sin{x})^2}\right]???

???y'=8\left(\frac{3}{4}\right)^7\frac{x^{28}}{\sin^7{x}}\left[\frac{192x^3\sin{x}-48x^4\cos{x}}{64\sin^2{x}}\right]???

???y'=\left(\frac{3}{4}\right)^7\frac{x^{28}}{\sin^7{x}}\left[\frac{24x^3\sin{x}-6x^4\cos{x}}{\sin^2{x}}\right]???

???y'=\left(\frac{3}{4}\right)^7\left[\frac{6x^{31}(4\sin{x}-x\cos{x})}{\sin^9{x}}\right]???


Let’s look at the combination of chain rule and quotient rule in a different way.


Example

Use chain rule to find the derivative.

???y=\frac{\left(6x^4-5\right)^2}{\left(7x^2+3\right)^3}???

In this case we want to do a double substitution, where

???u=6x^4-5???

???u'=24x^3???

and

???v=7x^2+3???

???v'=14x???

Our original equation becomes

???y=\frac{(u)^2}{(v)^3}???

and applying power rule, quotient rule, and chain rule together, the derivative is

???y'=\frac{(2u)(u')(v)^3-(u)^2\left(3v^2\right)(v')}{\left[(v)^3\right]^2}???

Back-substituting for ???u???, ???u'???, ???v??? and ???v'???, we get

???y'=\frac{2\left(6x^4-5\right)\left(24x^3\right)\left(7x^2+3\right)^3-\left(6x^4-5\right)^2\left[3\left(7x^2+3\right)^2(14x)\right]}{\left[\left(7x^2+3\right)^3\right]^2}???

???y'=\frac{48x^3\left(6x^4-5\right)\left(7x^2+3\right)^3-42x\left(6x^4-5\right)^2\left(7x^2+3\right)^2}{\left(7x^2+3\right)^6}???

???y'=\frac{6x\left(6x^4-5\right)\left[8x^2\left(7x^2+3\right)-7\left(6x^4-5\right)\right]}{\left(7x^2+3\right)^4}???

???y'=\frac{6x\left(6x^4-5\right)\left(56x^4+24x^2-42x^4+35\right)}{\left(7x^2+3\right)^4}???

???y'=\frac{6x\left(6x^4-5\right)\left(14x^4+24x^2+35\right)}{\left(7x^2+3\right)^4}???

 
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