Find the derivation of each of the following from the first principle:

Solution:

Let f(x) = ax + b

We need to find the derivative of f(x) i.e. f’(x)

We know that

$f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ …(i)

$f(x)=a x+b$

$f(x+h)=a(x+h)+b$

= ax + ah + b

Putting values in (i), we get

$\mathrm{f}^{\prime}(\mathrm{x})=\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{ax}+\mathrm{ah}+\mathrm{b}-(\mathrm{ax}+\mathrm{b})}{\mathrm{h}}$

$=\lim _{h \rightarrow 0} \frac{a x+a h+b-a x-b}{h}$

$=\lim _{h \rightarrow 0} \frac{a h}{h}$

$=\lim _{h \rightarrow 0} a$

$f^{\prime}(x)=a$

Hence, f’(x) = a