How do you derive a function composed of a division and a multiplication? Do you use the quotient rule or the product rule? Ex: $ f\left( x \right)={{\left( \dfrac{{{x}^{2}}-5{{x}^{-1}}}{\left( 5x \right)\left( -.5x \right)} \right)}^{3}}\left( 7{{x}^{6}}+4{{x}^{6}}-5{{x}^{2}} \right)$ ?

Hint: For this question, first you need to separate the equation into two parts. Then you need to find the derivative of the first part using the quotient rule of differentiation. Then you can use the product rule to find the derivative of the total question.

Complete step by step answer:
According to the problem, we are asked to derive a function composed of a division and a multiplication that is $ f\left( x \right)={{\left( \dfrac{{{x}^{2}}-5{{x}^{-1}}}{\left( 5x \right)\left( -.5x \right)} \right)}^{3}}\left( 7{{x}^{6}}+4{{x}^{6}}-5{{x}^{2}} \right)$.
We number this equation 1.
$ f\left( x \right)={{\left( \dfrac{{{x}^{2}}-5{{x}^{-1}}}{\left( 5x \right)\left( -.5x \right)} \right)}^{3}}\left( 11{{x}^{6}}-5{{x}^{2}} \right)$------ (1)
Now we divide it into two parts. The first part is $ {{\left( \dfrac{{{x}^{2}}-5{{x}^{-1}}}{\left( 5x \right)\left( -.5x \right)} \right)}^{3}}$ and the second part is $ \left( 7{{x}^{6}}+4{{x}^{6}}-5{{x}^{2}} \right)$.
Now we find the derivative of the first part using the quotient rule:
$ \Rightarrow \dfrac{d{{\left( \dfrac{{{x}^{2}}-5{{x}^{-1}}}{\left( 5x \right)\left( -.5x \right)} \right)}^{3}}}{dx}=3{{\left( \dfrac{{{x}^{2}}-5{{x}^{-1}}}{\left( 5x \right)\left( -.5x \right)} \right)}^{2}}\dfrac{d{{\left( \dfrac{{{x}^{2}}-5{{x}^{-1}}}{\left( 5x \right)\left( -.5x \right)} \right)}^{2}}}{dx}$
$ \Rightarrow \dfrac{d{{\left( \dfrac{{{x}^{2}}-5{{x}^{-1}}}{\left( 5x \right)\left( -.5x \right)} \right)}^{3}}}{dx}=3{{\left( \dfrac{{{x}^{2}}-5{{x}^{-1}}}{\left( 5x \right)\left( -.5x \right)} \right)}^{2}}\dfrac{d\left( \dfrac{-{{x}^{2}}+{{x}^{-1}}}{2.5{{x}^{2}}} \right)}{dx}$
Now, we use the quotient rule in the differentiation. Therefore we get,
$ \Rightarrow \dfrac{d{{\left( \dfrac{{{x}^{2}}-5{{x}^{-1}}}{\left( 5x \right)\left( -.5x \right)} \right)}^{3}}}{dx}=3{{\left( \dfrac{{{x}^{2}}-5{{x}^{-1}}}{\left( 5x \right)\left( -.5x \right)} \right)}^{2}}\left( -\left( \dfrac{2.5{{x}^{2}}\left( 2x+\dfrac{1}{{{x}^{2}}} \right)-5x\left( {{x}^{2}}-\dfrac{1}{x} \right)}{6.25{{x}^{4}}} \right) \right)$
$ \Rightarrow \dfrac{d{{\left( \dfrac{{{x}^{2}}-5{{x}^{-1}}}{\left( 5x \right)\left( -.5x \right)} \right)}^{3}}}{dx}=3{{\left( \dfrac{{{x}^{2}}-5{{x}^{-1}}}{\left( 5x \right)\left( -.5x \right)} \right)}^{2}}\left( -\left( \dfrac{5{{x}^{3}}+2.5-5{{x}^{2}}+5}{6.25{{x}^{4}}} \right) \right)$
$ \Rightarrow \dfrac{d{{\left( \dfrac{{{x}^{2}}-5{{x}^{-1}}}{\left( 5x \right)\left( -.5x \right)} \right)}^{3}}}{dx}=3{{\left( \dfrac{{{x}^{2}}-5{{x}^{-1}}}{\left( 5x \right)\left( -.5x \right)} \right)}^{2}}\left( -\dfrac{7.5}{6.25{{x}^{4}}} \right)$
$ \Rightarrow \dfrac{d{{\left( \dfrac{{{x}^{2}}-5{{x}^{-1}}}{\left( 5x \right)\left( -.5x \right)} \right)}^{3}}}{dx}=3{{\left( \dfrac{{{x}^{2}}-5{{x}^{-1}}}{\left( 5x \right)\left( -.5x \right)} \right)}^{2}}\left( -\dfrac{1.2}{{{x}^{4}}} \right)$----- (2)
$ \Rightarrow f'\left( x \right)=\left( 11{{x}^{6}}-5{{x}^{2}} \right)\left( 3{{\left( \dfrac{{{x}^{2}}-5{{x}^{-1}}}{\left( 5x \right)\left( -.5x \right)} \right)}^{2}}\left( -\dfrac{1.2}{{{x}^{4}}} \right) \right)+{{\left( \dfrac{{{x}^{2}}-5{{x}^{-1}}}{\left( 5x \right)\left( -.5x \right)} \right)}^{3}}\left( 66{{x}^{5}}-10x \right)$--- final answer.
Therefore, we have derived the derivative of a function composed of a division and a multiplication that is $ f\left( x \right)={{\left( \dfrac{{{x}^{2}}-5{{x}^{-1}}}{\left( 5x \right)\left( -.5x \right)} \right)}^{3}}\left( 7{{x}^{6}}+4{{x}^{6}}-5{{x}^{2}} \right)$.
Therefore, after all the derivation, we get the derivative of a function composed of a division and a multiplication that is $ f\left( x \right)={{\left( \dfrac{{{x}^{2}}-5{{x}^{-1}}}{\left( 5x \right)\left( -.5x \right)} \right)}^{3}}\left( 7{{x}^{6}}+4{{x}^{6}}-5{{x}^{2}} \right)$ as $ f'\left( x \right)=\left( 11{{x}^{6}}-5{{x}^{2}} \right)\left( 3{{\left( \dfrac{{{x}^{2}}-5{{x}^{-1}}}{\left( 5x \right)\left( -.5x \right)} \right)}^{2}}\left( -\dfrac{1.2}{{{x}^{4}}} \right) \right)+{{\left( \dfrac{{{x}^{2}}-5{{x}^{-1}}}{\left( 5x \right)\left( -.5x \right)} \right)}^{3}}\left( 66{{x}^{5}}-10x \right)$.

Note:
While doing the questions of this type, we need the derivatives of the basic polynomials. Also, we should be careful while doing all the substitutions. The derivatives of all the polynomials can be checked by using the integration.