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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.


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')\,

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