Hi I dont have an actual question but if I wanted to how can i differentiate a function like y=3^(x) with respect to x?

thanks

Alternatively, transform it into a function of \(\displaystyle e\).

\(\displaystyle y = 3^x\)

\(\displaystyle = e^{\ln{3^x}}\)

\(\displaystyle = e^{x\ln{3}}\)

\(\displaystyle = (e^x)^{\ln{3}}\).

Let \(\displaystyle u = e^x\) so that \(\displaystyle y = u^{\ln{3}}\).

\(\displaystyle \frac{du}{dx} = e^x\)

\(\displaystyle \frac{dy}{du} = (\ln{3})u^{\ln{3} - 1}\)

\(\displaystyle = (\ln{3})(e^x)^{\ln{3} - 1}\)

\(\displaystyle = (\ln{3})e^{x\ln{3} - x}\)

\(\displaystyle = (\ln{3})(e^{x\ln{3}})(e^{-x})\)

\(\displaystyle = (\ln{3})(e^{\ln{3^x}})(e^{-x})\)

\(\displaystyle = (\ln{3})(3^x)(e^{-x})\).

Therefore

\(\displaystyle \frac{dy}{dx} = \frac{du}{dx}\,\frac{dy}{du}\)

\(\displaystyle = e^x(\ln{3})(3^x)(e^{-x})\)

\(\displaystyle = (\ln{3})3^x\).