Question

find $\frac{dy}{dx}$. $y = 6(cos x+sin x)(cos x - sin x)$ $\frac{dy}{dx}=square$

Explanation:

Step1: Simplify the function

Use the difference - of - squares formula $(a + b)(a - b)=a^{2}-b^{2}$. Here $a=\cos x$ and $b = \sin x$, so $y = 6(\cos^{2}x-\sin^{2}x)$. By the double - angle formula $\cos^{2}x-\sin^{2}x=\cos2x$, then $y = 6\cos2x$.

Step2: Apply the chain rule

The derivative of $\cos u$ with respect to $x$ is $-\sin u\cdot\frac{du}{dx}$. Let $u = 2x$, then $\frac{du}{dx}=2$. The derivative of $y = 6\cos2x$ with respect to $x$ is $\frac{dy}{dx}=6\times(-\sin2x)\times2$.

Answer:

$- 12\sin2x$