Question

What is the derivative of $x$?

Answer 1

Use the limit definition of derivative to find the derivative of $x$. Assume ‘h’ as the small change in x and hence find the small change in the function f (x) given as f(x + h). Now, according to the limit definition of derivative apply the formula: $f'\left( x \right)=\displaystyle \lim_{h \to 0}\left( \dfrac{f\left( x+h \right)-f\left( x \right)}{h} \right)$, substitute the value of the given functions and simplify the limit to get the answer.

Complete step by step solution:
Here we have been provided with the function $f\left( x \right)=x$ and we are asked to find its derivative. Let us use the limit definition of the derivative to get the answer.
We know that derivative of a function is defined as the rate of change of function. In other words we can say that it is the measure of change in the value of the function with respect to the change in the value of the variable on which it depends. This change is infinitesimally small, that means tending to zero. Mathematically we have,
$\Rightarrow f'\left( x \right)=\displaystyle \lim_{h \to 0}\left( \dfrac{f\left( x+h \right)-f\left( x \right)}{h} \right)$
In the above formula we have ‘h’ as the infinitesimally small change in the variable x.
Now, let us come to the question. We have the function $f\left( x \right)=x$, so substituting (x + h) in place of x in the function we get,
$\Rightarrow f\left( x+h \right)=x+h$
Now, substituting the value of the functions f (x) and f (x + h) in the limit formula we get,
$\begin{aligned} & \Rightarrow f'\left( x \right)=\displaystyle \lim_{h \to 0}\left( \dfrac{\left( x+h \right)-\left( x \right)}{h} \right) \\\ & \Rightarrow f'\left( x \right)=\displaystyle \lim_{h \to 0}\left( \dfrac{h}{h} \right) \\\ & \Rightarrow f'\left( x \right)=\displaystyle \lim_{h \to 0}1 \\\ \end{aligned}$
Substituting the value h = 0 we get,
$\Rightarrow f'\left( x \right)=1$

Hence, the derivative of $x$ is 1.

Note: One may note that the limit definition of derivative is also known as the first principle of differentiation. It is the basic definition of derivative and all the formulas of differentiation of different functions are derived from the first principle. We can also use the formula $\dfrac{d\left[ {{x}^{n}} \right]}{dx}=n{{x}^{n-1}}$ for $n\ne 0$, to solve the question. In the above question we have n =1.