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.