g(x)^f(x)

Function

context $ f,g : {\mathbb R} \to {\mathbb R} $
definition $ x\mapsto g(x)^{f(x)}:??$

Theorem

$\dfrac{{\mathrm d}}{{\mathrm d}x} g(x)^{f(x)} = \left[f(x)\,g'(x) + f'(x)\,g(x)\cdot \log\left(g(x)\right)\right]\cdot g(x)^{f(x)-1} $

Special cases

$\dfrac{{\mathrm d}}{{\mathrm d}x} x^c = c\cdot x^{c-1}$

$\dfrac{{\mathrm d}}{{\mathrm d}x} c^x = \log(c) \cdot c^x$

Reference


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