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If you measure the slope along every point of a smooth sin⁑(x)\sin(x) wave, those slope values trace out an exact cos⁑(x)\cos(x) wave. Why does the rate of change of a wave turn into another wave shifted by a quarter cycle?
To prove this connection without guessing, we have to rebuild the derivative directly from its fundamental definition.

The First Principles Foundation

The derivative from first principles finds the exact slope of a tangent line by bringing two points on a curve infinitely close together using the limit of a difference quotient:
fβ€²(x)=lim⁑hβ†’0f(x+h)βˆ’f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
Before we plug trigonometric functions into this limit, we need to unlock two special limits that govern tiny angles. What happens to sin⁑(h)\sin(h) and cos⁑(h)\cos(h) when hh shrinks toward zero?

Two Standard Trig Limits

When an angle hh is measured in radians, the arc length on a unit circle is practically identical to the vertical opposite side sin⁑(h)\sin(h) for tiny angles. In 1748, Leonhard Euler formalized this geometric insight into two fundamental limits:
lim⁑hβ†’0sin⁑(h)h=1andlim⁑hβ†’0cos⁑(h)βˆ’1h=0\lim_{h \to 0} \frac{\sin(h)}{h} = 1 \quad \text{and} \quad \lim_{h \to 0} \frac{\cos(h) - 1}{h} = 0
With these limits ready, how do we split up sin⁑(x+h)\sin(x+h) when it appears inside our difference quotient?

Differentiating sin⁑(x)\sin(x)