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प्रश्न
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
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उत्तर
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
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