Derivatives – Substitution – Vectoreverything

Substitution is a technique making it possible to differentiate some functions which otherwise could not be differentuated.

As an example below, y=sin(2x).

let u=2x because we can differentiate si n(u) with respect to u.

$\dfrac{dy}{dx} = \dfrac{dy}{du} \dfrac {du}{dx}$

$\dfrac {dy}{du} = cos(u)$

$\dfrac {du}{dx} = 2$

$\text{[math]}$ \dfrac{dy}{dx} = cos(u) 2 $\text{[math]}$

We move the 2 to the front and replace the u with 2x.

$\text{[math]}$ \dfrac{dy}{dx} = 2 cos(2x) $\text{[math]}$