Question

find y. y = 7x cos x y = □

Explanation:

Step1: Apply product - rule for first - derivative

The product rule is $(uv)' = u'v+uv'$, where $u = 7x$ and $v=\cos x$. So $y'=(7x)'\cos x+7x(\cos x)'$. Since $(7x)' = 7$ and $(\cos x)'=-\sin x$, we have $y' = 7\cos x-7x\sin x$.

Step2: Apply product - rule for second - derivative

For $y' = 7\cos x-7x\sin x$, find the derivative of each term. The derivative of $7\cos x$ is $- 7\sin x$. For the second term $-7x\sin x$, using the product rule with $u=-7x$ and $v = \sin x$, we have $(-7x\sin x)'=(-7x)'\sin x+(-7x)(\sin x)'=-7\sin x-7x\cos x$. Then $y''=-7\sin x-7\sin x - 7x\cos x=-14\sin x-7x\cos x$.

Answer:

$-14\sin x - 7x\cos x$