# 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.

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

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

Dividing both sides by $\Delta x$, we get

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

Example: Find the derivative of

We have the given function as

Differentiation with respect to variable $x$, we get

Now using the quotient rule of derivative, we have