Question

1. let $v(x)=xsin(x)$. find $v(x),v(x)$ and $v^{(3)}(x)$.

Explanation:

Step1: Find the first - derivative using product rule

The product rule states that if $y = u(x)v(x)$, then $y^\prime=u^\prime(x)v(x)+u(x)v^\prime(x)$. Here, $u = x$ and $v=\sin(x)$. So $u^\prime = 1$ and $v^\prime=\cos(x)$. Then $v^\prime(x)=1\times\sin(x)+x\times\cos(x)=\sin(x)+x\cos(x)$.

Step2: Find the second - derivative

Differentiate $v^\prime(x)=\sin(x)+x\cos(x)$ again. The derivative of $\sin(x)$ is $\cos(x)$, and for $x\cos(x)$ using the product rule with $u = x$ and $v=\cos(x)$ (where $u^\prime = 1$ and $v^\prime=-\sin(x)$), we have $v^{\prime\prime}(x)=\cos(x)+(\cos(x)-x\sin(x)) = 2\cos(x)-x\sin(x)$.

Step3: Find the third - derivative

Differentiate $v^{\prime\prime}(x)=2\cos(x)-x\sin(x)$. The derivative of $2\cos(x)$ is $- 2\sin(x)$, and for $x\sin(x)$ using the product rule again, we get $v^{(3)}(x)=-2\sin(x)-(\sin(x)+x\cos(x))=-3\sin(x)-x\cos(x)$.

Answer:

$v^\prime(x)=\sin(x)+x\cos(x)$
$v^{\prime\prime}(x)=2\cos(x)-x\sin(x)$
$v^{(3)}(x)=-3\sin(x)-x\cos(x)$