# What is the derivative of m(x)= (5^x)(x^7)?

May 1, 2016

$m ' \left(x\right) = {5}^{x} {x}^{6} \left(7 + x \ln 5\right)$

#### Explanation:

Using the rule for the product,

$m ' \left(x\right) = \left({5}^{x}\right) \left({x}^{7}\right) ' + \left({x}^{7}\right) \left({5}^{x}\right) '$.

$= 7 {x}^{6} {5}^{x} + {x}^{7} \left({e}^{x \ln 5}\right) '$

$= 7 {x}^{6} {5}^{x} + {x}^{7} \left({e}^{x \ln 5}\right) \ln 5$, using function of function rule.

$= 7 {x}^{6} {5}^{x} + {x}^{7} \left({5}^{x}\right) \ln 5$.

$= {5}^{x} {x}^{6} \left(7 + x \ln 5\right)$