Question

How do I find the derivative of $\dfrac{1}{x}$ using the difference quotient?

Answer 1

Hint: We start solving the problem by representing the given equation with a function. We then apply the difference quotient method as the derivative of a function $f\left( x \right)$ as $\displaystyle \lim_{h \to 0}\dfrac{f\left( x+h \right)-f\left( x \right)}{h}$ to proceed through the problem. We then make the necessary calculations and make use of the fact that $\displaystyle \lim_{x\to a}f\left( x \right)=f\left( a \right)$ to proceed further through the problem. We then make the necessary calculations to get the required result of derivative of the given function.

Complete step-by-step answer:
According to the problem, we are asked to find the derivative of $\dfrac{1}{x}$ using the difference quotient.
Let us assume $f\left( x \right)=\dfrac{1}{x}$ ---(1).
From difference quotient method, we know that the derivative of a function $f\left( x \right)$ is defined as $\displaystyle \lim_{h \to 0}\dfrac{f\left( x+h \right)-f\left( x \right)}{h}$. Let us use this result in equation (1).
So, we have ${{f}^{'}}\left( x \right)=\displaystyle \lim_{h \to 0}\dfrac{\dfrac{1}{x+h}-\dfrac{1}{x}}{h}$.
$\Rightarrow {{f}^{'}}\left( x \right)=\displaystyle \lim_{h \to 0}\dfrac{\dfrac{x-\left( x+h \right)}{x\left( x+h \right)}}{h}$.
$\Rightarrow {{f}^{'}}\left( x \right)=\displaystyle \lim_{h \to 0}\dfrac{x-x-h}{xh\left( x+h \right)}$.
$\Rightarrow {{f}^{'}}\left( x \right)=\displaystyle \lim_{h \to 0}\dfrac{-h}{xh\left( x+h \right)}$.
$\Rightarrow {{f}^{'}}\left( x \right)=\displaystyle \lim_{h \to 0}\dfrac{-1}{x\left( x+h \right)}$ ---(3).
We know that $\displaystyle \lim_{x\to a}f\left( x \right)=f\left( a \right)$. Let us use this result in equation (3).
$\Rightarrow {{f}^{'}}\left( x \right)=\dfrac{-1}{x\left( x+0 \right)}$.
$\Rightarrow {{f}^{'}}\left( x \right)=\dfrac{-1}{x\left( x \right)}$.
$\Rightarrow {{f}^{'}}\left( x \right)=\dfrac{-1}{{{x}^{2}}}$.
So, we have found the derivative of the function $\dfrac{1}{x}$ using difference quotient method as $\dfrac{-1}{{{x}^{2}}}$.
$\therefore $ The derivative of the function $\dfrac{1}{x}$ using a difference quotient method is $\dfrac{-1}{{{x}^{2}}}$.

Note: We should perform each step carefully in order to avoid confusion and calculation mistakes while solving this problem. Whenever we get this type of problem, we first try to recall the required definition to get the required answer. Similarly, we can expect problems to find the derivative of the function $\sin x$ using the difference quotient formula.