What is the derivative of $π^{x}$ $pi^x$ ?

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Answer 1 · 2016-06-26T05:06:09

$\frac{d}{d x} π^{x} = π^{x} ln (π)$ $d/dxpi^x = pi^xln(pi)$

$\frac{d}{d x} π^{x} = \frac{d}{d x} e^{ln (π^{x})}$ $d/dxpi^x = d/dx e^ln(pi^x)$

$= \frac{d}{d x} e^{x ln (π)}$ $=d/dxe^(xln(pi))$

$= e^{x ln (π)} (\frac{d}{d x} x ln (π))$ $=e^(xln(pi))(d/dxxln(pi))$

(By applying the chain rule with the functions $e^{x}$ $e^x$ and $x ln (π)$ $xln(pi)$ )

$= e^{ln (π^{x})} ln (π)$ $=e^ln(pi^x)ln(pi)$

$= π^{x} ln (π)$ $=pi^xln(pi)$

Note that this method can be generalized to show that $\frac{d}{d x} a^{x} = a^{x} ln (a)$ $d/dxa^x = a^xln(a)$ for any constant $a > 0$ $a>0$

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