Quotient Rule of Differentiation

The quotient rule is a formula for taking the derivative of a quotient of two functions. This formula makes it somewhat easier to keep track of all of the terms.
If $u(x)$ and $v(x)$ are two functions of $x$ and $\displaystyle f(x)=\dfrac{u(x)}{v(x)}$, then
$$f'(x)=\dfrac{u'(x)v(x)-u(x)v'(x)}{v(x)^2}$$ Expressions like $\displaystyle \dfrac{x^2+x+1}{4x-2}$, $\displaystyle \dfrac{\sqrt{x+2}}{x^2-4}$ and $\displaystyle \dfrac{x^4}{(x^3-x^2-1)^5}$ are called quotients because they represent the division of one function by another.

Found some students prefer to apply the product rule instead of applying the quotient rule. For instance, $\displaystyle \dfrac{x}{x+1}$ can be changed to $x(x+1)^{-1}$, but this is not a great idea to differentiate as many students made careless mistakes while applying product rule due to taking longer steps. Form of quotient must apply the quotient rule properly, if you want to reduce silly mistakes!

Example 1

Use the quotient rule to differentiate $\displaystyle f(x)=\dfrac{x}{x+1}$.

Example 2

Use the quotient rule to differentiate $\displaystyle f(x)=\dfrac{3x+1}{1-x}$.

Example 3

Use the quotient rule to differentiate $\displaystyle f(x)=\dfrac{3x+1}{1-x}$.

Example 4

Use the quotient rule to differentiate $\displaystyle f(x)=\dfrac{(2x+1)^3}{(4x-1)^4}$.

Extension Examples

These Extension Examples require to have some prerequisite skills including;
\( \begin{align} \displaystyle \dfrac{d}{dx}\sin{x} &= \cos{x} \\ \dfrac{d}{dx}\cos{x} &= -\sin{x} \\ \dfrac{d}{dx}e^x &= e^x \\ \dfrac{d}{dx}\log_e{x} &= \dfrac{1}{x} \\ \end{align} \)

Example 5

Find $\displaystyle \dfrac{dy}{dx}$ of $\displaystyle y=\dfrac{\sin{x}}{\cos{x}}$, known that $\dfrac{d}{dx}\sin{x} = \cos{x}$ and $\dfrac{d}{dx}\cos{x} = -\sin{x}$.

Example 6

Find $\displaystyle \dfrac{dy}{dx}$ of $\displaystyle y=\dfrac{e^x}{\log_e{x}}$, known that $\dfrac{d}{dx}e^x = e^x$ and $\dfrac{d}{dx}\log_e{x} = \dfrac{1}{x}$.