The new material here is just a list of formulas for taking derivatives of exponential, logarithm, trigonometric, and inverse trigonometric functions. Then any function made by composing these with polynomials or with each other can be differentiated by using the chain rule, product rule, etc. (These new formulas are not easy to derive, but we don't have to worry about that).

The first two are the essentials for exponential and logarithms: \begin{align*} {d\over dx}\;e^x&=e^x\\ {d\over dx}\;\ln\,x&={1\over x} \end{align*}

The next three are essential for trig functions: \begin{align*} {d\over dx}\;\sin\,x&=\cos\,x\\ {d\over dx}\;\cos\,x&=-\sin\,x\\ {d\over dx}\;\tan\,x&=\sec^2\,x \end{align*}

The next three are essential for inverse trig functions \begin{align*} {d\over dx}\;\arcsin\,x&={ 1 \over \sqrt{1-x^2 }}\\ {d\over dx}\;\arctan\,x&={1\over 1+x^2}\\ {d\over dx}\;\hbox{arcsec}\,x&={ 1 \over x\,\sqrt{x^2-1 }} \end{align*}

The previous formulas are the indispensable ones in practice, and are
the only ones that I personally remember (if I'm lucky). Other
formulas one *might* like to have seen are (with $a>0$ in the
first two):
\begin{align*}
{d\over dx}\;a^x&=\ln\,a\cdot a^x\\
{d\over dx}\;\log_a\,x&={1\over \ln\,a\cdot x}\\
{d\over dx}\;\sec\,x&=\tan\,x\,\sec\,x\\
{d\over dx}\;\csc\,x&=-\cot\,x\,\csc\,x\\
{d\over dx}\;\cot\,x&=-\csc^2\,x\\
{d\over dx}\;\arccos\,x&={ -1 \over \sqrt{1-x^2 }}\\
{d\over dx}\;\hbox{arccot}\,x&={-1\over 1+x^2} \\
{d\over dx}\;\hbox{arccsc}\,x&={ -1 \over x\,\sqrt{x^2-1 }}
\end{align*}

*(There are always some difficulties in figuring out which of the
infinitely-many possibilities to take for the values of the inverse
trig functions, and this is especially bad with* arccsc*, for
example. But we won't have time to worry about such things).*

To be able to use the
above formulas it is *not* necessary to know very many *other*
properties of these functions. For example, *it is not necessary
to be able to graph these functions to take their derivatives!*

#### Exercises

- Find ${ d \over dx }(e^{\cos x})$
- Find ${ d \over dx }(\arctan (2-e^x))$
- Find ${ d \over dx }(\sqrt{\ln\;(x-1)})$
- Find ${ d \over dx }(e^{2\cos x+5})$
- Find ${ d \over dx }(\arctan (1+\sin 2x))$
- Find ${d\over dx}\cos(e^x-x^2)$
- Find ${d\over dx}\sqrt[3]{1-\ln\,2x}$
- Find ${d\over dx}{e^x-1 \over e^x + 1}$
- Find ${ d \over dx }(\sqrt{\ln \;({ 1 \over x })})$