derivatives2.tex

Or: dealing with trig functions, exponential functions, and logarithms. I'll try to quickly recap relevant rules from algebra as they come up, but again, if this stuff is rusty, try looking at Khan Academy, PatrickJMT, or another similar resource.

\subsection{Normal Trig}

\subsection{Inverse Trig}

Remember that by using implicit differentiation we can find the derivative of inverse functions - and now we know about the derivatives of standard trig functions, so we can now find the derivatives of inverse trig functions (just what we all wanted).

\subsection{Exponentials and Logarithms}

\begin{equation*}
    \frac{d}{dx}\ln x = \frac{1}{x}
\end{equation*}

(We can actually prove the above from the derivative of $e^x$ - see if you can do it!)

\begin{equation*}
    \frac{d}{dx}\log_a x = \frac{1}{x \ln a}
\end{equation*}

The above can be proved from the derivative of $\ln x$ and the change of base formula for logarithms - again, try to do it!