15. if ( y = x sin x ), then ( \frac { d y } { d x } = )
16. if ( y = x ^ { 2 } ( e ^ { x } - 1 ) ), then ( \frac { d y } { d x } = )

Question

Accepted Answer

# Explanation:
## Step1: Use 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 $v=\sin x$. Then $u^\prime = 1$ and $v^\prime=\cos x$.
$$
\begin{align*}
\frac{dy}{dx}&=(x)^\prime\sin x+x(\sin x)^\prime\
\end{align*}
$$
## Step2: Calculate the derivatives
Since $(x)^\prime = 1$ and $(\sin x)^\prime=\cos x$, we substitute these values into the product - rule formula.
$$
\begin{align*}
\frac{dy}{dx}&=1\cdot\sin x+x\cdot\cos x\
&=\sin x + x\cos x
\end{align*}
$$

# Answer:
B. $\sin x + x\cos x$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Use the product rule

Step2: Calculate the derivatives

Answer: