(i) For a positive integer and base , prove that d dx [ _a !(x^ n ) ]= n x a , x>0.…

MATHEMATICS · Subjective · A Level · Mathematics · Exponentials and logarithms · Exponentials and logarithms

Problem

(i) For a positive integer

n

and base

a>0,\;a\neq1

, prove that

\frac{d}{dx}\bigl[\log_a\!(x^{n})\bigr]=\frac{n}{x\ln a}, \quad x>0.

(ii) Using the result from part (i), find the derivative of

\log_7\!(x^3)

x=2