User:Wolfmankurd/proofs

[edit] Differential of exponential

y=e^x

$\frac{dy}{dx}=\frac{e^{(x+h)}-e^x}{h}$

$\frac{dy}{dx}=\frac{e^x(e^h-1)}{h}$

Limit h → 0, (e^h --1)→ h

$\frac{dy}{dx}=\frac{e^xh}{h}$

$\frac{dy}{dx}=e^x$

y=sin(x)

$\frac{dy}{dx}=\frac{sin(x+h)-sin(x)}{h}$

$\frac{dy}{dx}=\frac{sin(x)cos(h)-sin(h)cos(x)-sin(x)}{h}$

Limit h → 0, sin(h)→ h, cos(h) → 1

$\frac{dy}{dx}=\frac{sin(x)1-hcos(x)-sin(x)}{h}$

$\frac{dy}{dx}=\frac{hcos(x)}{h}$

$\frac{dy}{dx}=cos(x)$