What is the derivative of 2^x?

Dec 14, 2015

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

Explanation:

Use the chain rule and the identity:

$\frac{d}{\mathrm{dt}} {e}^{t} = {e}^{t}$

Start by using properties of exponents:

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

So if we put $t = x \ln 2$, then:

$\frac{\mathrm{dt}}{\mathrm{dx}} = \ln 2$

and:

$\frac{d}{\mathrm{dx}} {2}^{x} = \frac{d}{\mathrm{dx}} {e}^{x \ln 2} = \frac{\mathrm{dt}}{\mathrm{dx}} \frac{d}{\mathrm{dt}} {e}^{t} = {e}^{t} \cdot \ln \left(2\right)$

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