Question

How do you find the derivative of $0$ using the limit definition?

Answer 1

Here, we have to find the derivative of the given function. We will use the limit definition formula for the given functions and then by substituting the limits, we will find the derivative of the function. Differentiation is a method of finding the derivative of the function or finding the rate of change of a function with respect to one variable.

Formula Used:
Limit definition is given by ${f^{'}}\left( x \right) = \mathop {\lim }\limits_{\Delta x \to 0} \dfrac{{f\left( {x + \Delta x} \right) - f\left( x \right)}}{{\Delta x}}$ .

Complete step by step solution:
We are given a term $0$.
Let the given function be $f\left( x \right)$.So, we get
$\Rightarrow f\left( x \right) = 0$ ;…………………………………………………………………………………………………………$\left( 1 \right)$
Now, we will find $f\left( {x + \Delta x} \right)$, so we get
$\Rightarrow f\left( {x + \Delta x} \right) = 0$ ……………………………………………………………….$\left( 2 \right)$
Now, we will find the derivative of $0$ using the limit definition.
Limit definition is given by ${f^{'}}\left( x \right) = \mathop {\lim }\limits_{\Delta x \to 0} \dfrac{{f\left( {x + \Delta x} \right) - f\left( x \right)}}{{\Delta x}}$ .
Now, by substituting the equation $\left( 1 \right)$ and equation $\left( 2 \right)$ in the limit definition, we get
$\Rightarrow {f^{'}}\left( x \right) = \mathop {\lim }\limits_{\Delta x \to 0} \dfrac{{0 - 0}}{{\Delta x}}$ .
Now, by simplifying the equation, we get
$\Rightarrow {f^{'}}\left( x \right) = \mathop {\lim }\limits_{\Delta x \to 0} \dfrac{0}{{\Delta x}}$ .
Now, by substituting the limits, we get
$\Rightarrow {f^{'}}\left( x \right) = \mathop {\lim }\limits_{\Delta x \to 0} \dfrac{0}{0}$ .
$\Rightarrow {f^{'}}\left( x \right) = 0$ .

Therefore, the derivative of $0$ using the limit definition is $0$.

Note:
We know that the reverse process of differentiation is called antidifferentiation. We should remember some rules in differentiation which include that the derivative of a constant multiplied by a function is equal to the constant multiplied by the derivative of the function. The derivative of a constant is always zero since zero is a constant; its derivative is zero. The derivative function is used to find the highest and the lowest point of the curve in a graph or to know its turning point. The derivative function is also used to find the tangent and normal to the curve.