Derivative of Sin X: Formula, Proofs, and Examples

The derivative of $\sin x$ is $\cos x$, which represents the rate of change of the sine function. Using first principles, ddx(sin⁡x)=lim⁡h→0sin⁡(x+h)−sin⁡xh=cos⁡x\frac{d}{dx}(\sin x) = \lim_{h \to 0} \frac{\sin(x + h) – \sin x}{h} = \cos xdxd​(sinx)=limh→0​hsin(x+h)−sinx​=cosx. This formula is fundamental in calculus for solving trigonometric problems. For example, if $y=\sin x$, then dydx=cos⁡x\frac{dy}{dx} = \cos xdxdy​=cosx. At $x=\pi /3$, $\cos (\pi /3)=1/2$. The derivative of $\sin x$ is essential in physics, engineering, and mathematics for analyzing periodic motion, wave functions, and oscillatory behavior.