Question

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

Short answer

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

Step-by-step solution

Given information

We are given constant and identity function

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

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

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

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