Quotient Rule of Derivatives

Product rule of derivative is \frac{d}{{dx}}\left[ {\frac{{f\left( x  \right)}}{{g\left( x \right)}}} \right] = \frac{{g\left( x \right)f'\left( x  \right) - f\left( x \right)g'\left( x \right)}}{{{{\left[ {g\left( x \right)}  \right]}^2}}}. In words we can read as derivative of quotient of two functions is equal to second function as it is and derivative of first function minus first function as it is and derivative of second function divided by square of second function. This product rule can be proving using first principle or derivative by definition.
Consider a function of the form y = \frac{{f\left( x \right)}}{{g\left(  x \right)}}.
First we take the increment or small change in the function.

\begin{gathered}y + \Delta y = \frac{{f\left( {x + \Delta x}  \right)}}{{g\left( {x + \Delta x} \right)}} \\ \Rightarrow \Delta y = \frac{{f\left( {x +  \Delta x} \right)}}{{g\left( {x + \Delta x} \right)}} - y \\ \end{gathered}


Putting the value of function y = \frac{{f\left( x \right)}}{{g\left( x \right)}} in the above equation, we get

\begin{gathered} \Rightarrow \Delta y = \frac{{f\left( {x +  \Delta x} \right)}}{{g\left( {x + \Delta x} \right)}} - \frac{{f\left( x  \right)}}{{g\left( x \right)}} \\ \Rightarrow \Delta y = \frac{{f\left( {x +  \Delta x} \right)g\left( x \right) - f\left( x \right)g\left( {x + \Delta x}  \right)}}{{g\left( {x + \Delta x} \right)g\left( x \right)}} \\ \end{gathered}


Subtracting and adding f\left(  x \right)g\left( x \right) on the right hand side, we have

\begin{gathered}\Rightarrow \Delta y = \frac{{f\left( {x +  \Delta x} \right)g\left( x \right) - f\left( x \right)g\left( x \right) +  f\left( x \right)g\left( x \right) - f\left( x \right)g\left( {x + \Delta x}  \right)}}{{g\left( {x + \Delta x} \right)g\left( x \right)}} \\ \Rightarrow \Delta y = \frac{{g\left( x  \right)\left[ {f\left( {x + \Delta x} \right) - f\left( x \right)} \right] +  f\left( x \right)\left[ {g\left( x \right) - g\left( {x + \Delta x} \right)}  \right]}}{{g\left( {x + \Delta x} \right)g\left( x \right)}} \\ \Rightarrow \Delta y = \frac{{g\left( x  \right)\left[ {f\left( {x + \Delta x} \right) - f\left( x \right)} \right] -  f\left( x \right)\left[ {g\left( {x + \Delta x} \right) - g\left( x \right)}  \right]}}{{g\left( {x + \Delta x} \right)g\left( x \right)}} \\ \end{gathered}


Dividing both sides by \Delta  x, we get

\begin{gathered}\Rightarrow \frac{{\Delta y}}{{\Delta x}} =  \frac{{g\left( x \right)\left[ {f\left( {x + \Delta x} \right) - f\left( x  \right)} \right] - f\left( x \right)\left[ {g\left( {x + \Delta x} \right) -  g\left( x \right)} \right]}}{{\Delta xg\left( {x + \Delta x} \right)g\left( x  \right)}} \\ \Rightarrow \frac{{\Delta y}}{{\Delta x}} =  \left[ {\frac{{f\left( {x + \Delta x} \right) - f\left( x \right)}}{{\Delta  x}}} \right]\frac{{g\left( x \right)}}{{g\left( {x + \Delta x} \right)g\left( x  \right)}} - \frac{{f\left( x \right)}}{{g\left( {x + \Delta x} \right)g\left( x  \right)}}\left[ {\frac{{g\left( {x + \Delta x} \right) - g\left( x  \right)}}{{\Delta x}}} \right] \\ \end{gathered}


Taking limit of both sides as \Delta x \to 0, we have

\begin{gathered}\Rightarrow \mathop {\lim }\limits_{\Delta x  \to 0} \frac{{\Delta y}}{{\Delta x}} = \mathop {\lim }\limits_{\Delta x \to 0}  \left[ {\frac{{f\left( {x + \Delta x} \right) - f\left( x \right)}}{{\Delta  x}}} \right]\mathop {\lim }\limits_{\Delta x \to 0} \frac{{g\left( x  \right)}}{{g\left( {x + \Delta x} \right)g\left( x \right)}} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - \mathop {\lim  }\limits_{\Delta x \to 0} \frac{{f\left( x \right)}}{{g\left( {x + \Delta x}  \right)g\left( x \right)}}\mathop {\lim }\limits_{\Delta x \to 0} \left[  {\frac{{g\left( {x + \Delta x} \right) - g\left( x \right)}}{{\Delta x}}}  \right] \\ \Rightarrow \frac{{dy}}{{dx}} = f'\left( x  \right)\frac{{g\left( x \right)}}{{g\left( {x + 0} \right)g\left( x \right)}} -  \frac{{f\left( x \right)}}{{g\left( {x + 0} \right)g\left( x \right)}}g'\left(  x \right) \\ \frac{{dy}}{{dx}} = \frac{{g\left( x  \right)f'\left( x \right) - f\left( x \right)g'\left( x \right)}}{{{{\left[  {g\left( x \right)} \right]}^2}}} \\ \frac{d}{{dx}}\left[ {\frac{{f\left( x  \right)}}{{g\left( x \right)}}} \right] = \frac{{g\left( x \right)f'\left( x  \right) - f\left( x \right)g'\left( x \right)}}{{{{\left[ {g\left( x \right)}  \right]}^2}}} \\ \end{gathered}

Example: Find the derivative of

y = \frac{{{x^3} -  8}}{{{x^3} + 8}}


We have the given function as

y =  \frac{{{x^3} - 8}}{{{x^3} + 8}}


Differentiation with respect to variable x, we get

\frac{{dy}}{{dx}}  = \frac{d}{{dx}}\left( {\frac{{{x^3} - 8}}{{{x^3} + 8}}} \right)


Now using the quotient rule of derivative, we have

\begin{gathered}\frac{{dy}}{{dx}} = \frac{{\left( {{x^3} + 8}  \right)\frac{d}{{dx}}\left( {{x^3} - 8} \right) - \left( {{x^3} - 8}  \right)\frac{d}{{dx}}\left( {{x^3} + 8} \right)}}{{{{\left( {{x^3} + 8}  \right)}^2}}} \\ \frac{{dy}}{{dx}} = \frac{{\left( {{x^3} + 8}  \right)3{x^2} - \left( {{x^3} - 8} \right)3{x^2}}}{{{{\left( {{x^3} + 8}  \right)}^2}}} \\ \frac{{dy}}{{dx}} = \frac{{\left( {3{x^5} +  24{x^2}} \right) - \left( {3{x^5} - 24{x^2}} \right)}}{{{{\left( {{x^3} + 8}  \right)}^2}}} \\ \frac{{dy}}{{dx}} =  \frac{{48{x^2}}}{{{{\left( {{x^3} + 8} \right)}^2}}} \\ \end{gathered}

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