Math Insight

Derivatives of transcendental functions

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

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