Chapter 3 - Derivatives - 3.5 Derivatives of Trigonometric Functions - 3.5 Exercises - Page 170: 72

$2\cos 2x$

We are given the function: $f(x)=\sin 2x$ Determine the derivative of $f(x)$. First use the identity $\sin 2x=2\sin x\cos x$ $\dfrac{d}{dx}(\sin 2x)=\dfrac{d}{dx}(2\sin x\cos x)=2[(\sin x)'\cos x+\sin x \cos'x]$ $=2[\cos x\cos x+\sin x(-\sin x)]=2(\cos^2 x-\sin^2 x)$ Apply the identity: $\cos 2x=\cos^2 x-\sin^2 x$ $\dfrac{d}{dx}(\sin 2x)=2\cos 2x$