Question

let $g(x)=x\cos(x)$. find $g(x)$. choose 1 answer: a $-\sin(x)$ b $\cos(x)-x\sin(x)$ c $\cos(x)+x\sin(x)$ d $-\cos(x)$

Explanation:

Step1: Apply the product rule

The product rule states that if \(y = u\cdot v\), then \(y^\prime=u^\prime v + uv^\prime\). Let \(u = x\) and \(v=\cos(x)\).

Step2: Find \(u^\prime\) and \(v^\prime\)

The derivative of \(u = x\) is \(u^\prime=\frac{d}{dx}(x)=1\). The derivative of \(v=\cos(x)\) is \(v^\prime=\frac{d}{dx}(\cos(x))=-\sin(x)\).

Step3: Substitute into the product rule formula

\(g^\prime(x)=u^\prime v+uv^\prime=(1)\cdot\cos(x)+x\cdot(-\sin(x))\).

Answer:

B. \(\cos(x)-x\sin(x)\)