Question

Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero 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->0) (f(x + h) - f(x))/h`

= `lim_(h->0) ((ax + ah + b)^n - (ax + b)^n)/h`

= `lim_(h->0) ((ax + b)^n (1 + (ah)/(ax + b))^n - (ax + b)^n)/h`

= `(ax + b)^n lim_(h->0)((1 + (ah)/(ax + b))^n - 1)/h`

= `(ax + b)^n lim_(h->0) 1/h [{1 + n}((ah)/(ax + b)) + (n(n - 1))/2 ((ah)/(ax + b))^2 + ...}-1]`    (Using binomial theorem)

= `(ax + b)^n lim_(h->0)1/h [n ((ah)/(ax + b)) + (n (n - 1)a^2h^2)/(2(ax + b)^2] + ("Terms containing higher degrees of h"))]`

= `(ax + b)^n lim_(h->0) [(na)/(ax + b) + (n(n + 1)a^2 h)/(2 (ax + b))^2 + ...]`

= `(ax + b)^n [(na)/(ax + b) + 0]`

= `na(ax + b)^n/((ax + b))`

= na (ax + b)^{n - 1}