Calc2.1

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

We have derivative formulas for functions such as e^{x}\, and x^{3}\, but not for a function involving x\, both in the base and the exponent. However, any problem with x\, of some form in the base and exponent can be solved by logarithmic differentiation. The process is simple. Take the logarithm of both sides and the exponent drops down by the laws of logarithms.

\ln f(x)=x\ln x\,

This is a function we can derive

{\frac  {1}{f(x)}}f'(x)=\ln x+x{\frac  {1}{x}}=\ln x+1\,

Now solve for f'(x)\, and then replace f(x)\, with its given value, x^{x}\,

f'(x)=f(x)(\ln x+1)=x^{x}(\ln x+1)\,


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