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
Element of
