Q. 84 - Calculus

Question

use the definition of derivative to directly prove the differentiation rules for constant and identity function

Step-by-Step Solution

Answer

We use definition of the derivative to prove the differentiation rules for constant and identity function

Step 1: Given information

We are given constant and identity function

Step 2: Derivative for constant function

Consider a constant function $f:R\to R$ such that $f\left(x\right)=c$ for all $x\in R$ 

Now we use definition of derivative

$$\begin{gathered} \underset{h\to 0}{\lim }\frac{f(x+h)-f\left(x\right)}{h} \\ \underset{h\to 0}{\lim }\frac{c-c}{h} as for any x we have f\left(x\right)=c \\ =0 \end{gathered}$$

Step 3: Derivative for identity function

Consider the identity function $f:R\to R$ given by $f\left(x\right)=x$

Now we apply the definition of the derivative

$$\begin{gathered} \underset{h\to 0}{\lim }\frac{f(x+h)-f\left(x\right)}{h} \\ \underset{h\to 0}{\lim }\frac{x+h-x}{h} \\ \underset{h\to 0}{\lim }\frac{h}{h} \\ =1 \end{gathered}$$