# How do you find the derivative of y = (ln 2)^x?

Apr 15, 2016

$\left(\ln \left(\ln 2\right)\right) {\left(\ln 2\right)}^{x}$

#### Explanation:

Use ${a}^{x} = {e}^{x \ln a} \mathmr{and} \frac{d}{\mathrm{dx}} \left({e}^{a x}\right) = a {e}^{a x}$.

Here, a=ln 2.

$y ' = \left({e}^{\left(\ln \left(\ln 2\right)\right) x}\right) ' = \ln \left(\ln 2\right) {e}^{\ln 2 x} = \ln \left(\ln 2\right) {\left(\ln 2\right)}^{x}$
ln 2 = 0.6931 and ln (ln 2)=-0.3665, nearly..

Interestingly, the nth derivative is ${\left(\ln \left(\ln 2\right)\right)}^{n} {\left(\ln 2\right)}^{x}$
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