Calc2.9

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Find the derivative of $f(x)\,$ with respect to $x\,$: $f(x)=4^{{\sin x}}\,$

Most calculus books will give a formula for this type of problem. For any problem where you have a constant base and an exponent that is a function of $x\,$, the derivative can be done by memorization. If

$f(x)=a^{{b(x)}}\,$ where $a\,$ is a constant and $b(x)\,$ is some function of $x\,$, the derivative is given by

$f'(x)=a^{{b(x)}}\ln(a)b'(x)\,$

To truly understand the problem though, it is better to know where the formula comes from. For this problem, take the logarithm of both sides. Then using the laws of logarithms, we know that $\ln c^{d}=d\ln c\,$. Thus, for this problem,

$\ln f(x)=\sin x\ln 4\,$

This puts it in a form we can easily integrate as it is simply $\sin x\,$ multiplied by a constant on the right.

${\frac {1}{f(x)}}f'(x)=\cos x\ln 4\,$

Now solving for $f'(x)\,$ and substituting in our original value of $f(x)\,$ we get

$f'(x)=4^{{\sin x}}\cos x\ln 4\,$