[Math] Chain rule of fractions

chain rulederivativesfunctions

How can I use chain rule to find the derivative of $y=\frac{(3x-3)^2}{x}$?
Should I multiply the derivative of whole $y$ with the derivative of $(3x-3)^2$ and then after that all with derivative of $(3x-3)$?

Best Answer

With stuff like this you can also expand it to $f(x)=9x-18+\frac 9x$ and derivate $f'(x)=9-\frac 9{x^2}$, this is more efficient.

However if you have calculus withdrawal symptoms already you can either use:

  • The product rule : $(uv)'=u'v+v'u$

$f'(x)=\underbrace{6(3x-3)}_{u'}\times \underbrace{\dfrac 1x}_{v} + \underbrace{(3x-3)^2}_{u}\times \underbrace{\dfrac{-1}{x^2}}_{v'}$

  • Or the quotient rule: $(\frac uv)'=\dfrac{u'v-v'u}{v^2}$

$f'(x)=\dfrac{\underbrace{6(3x-3)}_{u'}\times \underbrace{x}_{v}-\underbrace{1}_{v'}\times\underbrace{(3x-3)^2}_{u}}{\underbrace{x^2}_{v^2}}$

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