Why d/dx sin(x) = cos(x)

The visual intuition

Watch the tangent line sweep along sin(x). Its slope at every point is exactly the height of cos(x) at that same x — that's the whole claim.

The proof

Start from the limit definition of the derivative, apply the angle-addition identity, then use two classic limits:

$\lim_{h \to 0}\frac{\sin h}{h} = 1 \qquad \lim_{h \to 0}\frac{\cos h - 1}{h} = 0$

Substitute both, and what's left is $\cos x$. The video walks through each step.