Question

Find the derivative of the function.

Answer 1

\[(\frac{ 1 }{ 3 })^x\]

Answer 2

here we can apply this rule:
\[\huge \frac{{d{a^x}}}{{dx}} = {a^x}\ln a,\quad a > 0\]

Answer 3

what rule is that called?

Answer 4

I don't know. It is an identity, which can be derived, if we apply the definition of first derivative of a function

Answer 5

Using that rule, \[a=\frac{ 1 }{3 }\]
=\[(\frac{ 1 }{3 })^x \times(\frac{ 1 }{ 3 })\ln\]?

Answer 6

more precisely, we get this:
\[\Large \frac{d}{{dx}}{\left( {\frac{1}{3}} \right)^x} = {\left( {\frac{1}{3}} \right)^x}\ln \left( {\frac{1}{3}} \right) = - {3^{ - x}}\ln 3\]

Answer 7

Okay, got it. Thanks! :D

Answer 8

:)