# Differential calculus reminder

This is just an online note to myself about differential calculus. A level maths again…

Calculating $\frac{d}{dx}2^x$

$2^x = e^u$ where $u=\ln(2^x)$

Using the chain rule: $\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}$

$\frac{dy}{du} = e^u$ (as $\frac{d}{dx}e^x = e^x$)

$\ln(2^x) = x\ln(2)$ hence $\frac{d}{dx}\ln(2^x) = \ln(2)$.

So $\frac{d}{dx}2^x = e^{\ln(2^x)}\ln(2) = 2^x\ln(2)$

And $e$, Euler’s number, really is magical.