Calc2.2

From Exampleproblems

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

This is a general form of the previous problem. The same process will be followed and when we are finished we will have a formula which can be used to differentiate any function which involves a function of $x\,$ in the base and another in the exponent.

$f=g^h\,$

As before, take the natural log of both sides to change the function into a form of which we can take the derivative.

$\ln f=h\ln g\,$

$\frac{1}{f}f'=h'\ln g+\frac{h}{g}g'\,$

$f'=f(h'\ln g+\frac{h}{g}g')\,$

$f'=g^h(h'\ln g+\frac{h}{g}g')\,$