Question

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed nonzero constants and m and n are integers): (ax +b)n

Answer 1

Let f(x) = (ax +b)n. Accordingly, f(x+h) = {a(x+h) + b}n = (ax + ah +b )n
by first principle,
f'(x) = $\lim_{h\rightarrow 0}$ $\frac{f(x+h)-f(x)}{h}$
=$\lim_{h\rightarrow 0}$ (ax+ah+b)n -(ax+b)n
=$\lim_{h\rightarrow 0}$ $\frac{(ax+b)^n(1+\frac{ah}{ax+b})^n-(ax+b)^n}{h}$
=(ax+b)n \lim_{h\rightarrow 0}$$\frac{1}{n}[{ 1+n($\frac{ah}{ax+b}$) +$\frac{n(n-1)}{2}$($\frac{ah}{ax+b}$)2 + .....}-1](Using binomial theorem)
=(ax+b)n $\lim_{h\rightarrow 0}$ $\frac{1}{h}$[n($\frac{ah}{ax+b}$) + n(n-1)$\frac{a^2h^2}{(ax+b)^2}$ + .....(Terms containing higher degrees of h)]
=(ax+b)n$\lim_{h\rightarrow 0}$ $\frac{1}{h}$[ $\frac{na}{ax+b}$+ n(n-1)$\frac{a^2h^2}{(ax+b)^2}$ + ..]
=(ax+b)n [$\frac{na}{(ax+b)}$ + 0]
=na$\frac{(ax+b)^n}{(ax+b)}$
=na (ax+b)n-1