User:Wolfmankurd/proofs

Differential of exponential
y=ex

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

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

Limit h → 0, (eh --1)→ h

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

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

Differential of sin 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

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

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

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