2.2.211 AP Calculus Exam practice problem

[karush]

211(DOY)
If If $y=x \sin x,$ then $\dfrac{dy}{dx}=$
(A) $\sin x + \cos x$
(B) $\sin x + x \cos x$
(C) $\sin x + \cos x$
(D) $x(\sin x + \cos x)$
(E) $x(\sin x - \cos x)$
Solution

Spoiler

note: $y'=\dfrac{dy}{dx}$
Apply the Product Rule $$(f\cdot g)'=f\:'\cdot g+f\cdot g'$$
then
$$y'=x'\sin x+(\sin x)' x$$
simplify
$$y'=\sin x+x\cos x \quad (B)$$

ok this is a relatively simple problem but was wondering if $y'$ should be used in combination with $\dfrac{dy}{dx}=$
also any other typos ,, suggestion,, etc

 

[topsquark]

211(DOY)
If If $y=x \sin x,$ then $\dfrac{dy}{dx}=$
(A) $\sin x + \cos x$
(B) $\sin x + x \cos x$
(C) $\sin x + \cos x$
(D) $x(\sin x + \cos x)$
(E) $x(\sin x - \cos x)$
Solution

Spoiler

note: $y'=\dfrac{dy}{dx}$
Apply the Product Rule $$(f\cdot g)'=f\:'\cdot g+f\cdot g'$$
then
$$y'=x'\sin x+(\sin x)' x$$
simplify
$$y'=\sin x+x\cos x \quad (B)$$

ok this is a relatively simple problem but was wondering if $y'$ should be used in combination with $\dfrac{dy}{dx}=$
also any other typos ,, suggestion,, etc

It looks good. If you are given \(\displaystyle \dfrac{dy}{dx}\) in the problem statement then I would stick with that notation.

-Dan