$\begingroup$I need a reminder of how to go about taking the derivative of $c^{n^x}$ where $x$ and $c$ are constants. An example would be
$$\large 4^{n^2}$$
Thanks in advance.
$\endgroup$32 Answers
$\begingroup$You can write $4^{n^2} = e^{n^2\ln4}$ and then go at it with a chain rule, getting $(2n\ln{4})e^{n^2 \ln 4}$
$\endgroup$$\begingroup$You first differentiate the exponential using the rule $(a^f)'=\log a\,a^f f'$,
$$\left(a^{x^b}\right)'=\log a\,a^{x^b}\left({x^b}\right)'$$
then expand the derivative of the power,
$$\left(a^{x^b}\right)'=\log a\,a^{x^b}b\,x^{b-1}.$$
Alternatively, you can use the so-called logarithmic derivative, using the rule
$$(\log g)'=\frac{g'}g\iff g'=g\;(\log g)'.$$
Then
$$\left(a^{x^b}\right)'=a^{x^b}\left(\log a^{x^b}\right)'=a^{x^b}\left(x^b\log a\right)'=a^{x^b}b\,x^{b-1}\log a.$$
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