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Mathematics-Online course: Prepcourse Mathematics - Analysis - Differential Calculus

Logarithmic Derivation

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The formula

$\displaystyle f'(x) = f(x) \frac{d}{dx}\ln\vert f(x)\vert $

can be used to determine the derivative of functions $ y = g(x)^{h(x)}$ with $ g(x)>0$ . One obtains

$\displaystyle \frac{d y}{dx}=g(x)^{h(x)}\frac{d}{dx}\big( h(x) \ln g(x) \big) \;. $

(Authors: App/Höllig/Abele )
For

$\displaystyle f(x)=x^x$

with $ x>0$, the derivative calculates as

$\displaystyle f'(x)=x^x\frac{d}{dx}\ln(x^x)= x^x \frac{d}{dx}\,( x\ln x) = x^x(\ln x +1)\,.$

For $ g(x) = x^{\ln x}$ it is

$\displaystyle g'(x) = x^{\ln x} \frac{d}{dx}(\ln x)^2 = 2x^{\ln x} \frac{\ln x}{x}\,. $

(Authors: App/Höllig/Abele )
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  automatically generated 9/18/2007