What is the derivative of $x\sin x$? Ok so I know the answer of $\frac{d}{dx}x\sin(x) = \sin(x)+ x\cos(x)$...but how exactly do you get there? I know $\frac{d}{dx} \sin{x} = \cos{x}$. But where does the additional $\sin(x)$ (in the answer) come in? 
 A: Hint: Use the product rule for derivatives.
Alternately, proceed by definition, using trig properties (continuity, angle sum):
$$\begin{align}\frac{d}{dx}[x\sin x] &= \lim_{h\to 0}\frac{(x+h)\sin(x+h)-x\sin x}h\\ &= \lim_{h\to 0}\frac{h\sin(x+h)}h+x\lim_{h\to 0}\frac{\sin(x+h)-\sin x}h\\ &= \lim_{h\to 0}\sin(x+h)+x\lim_{h\to 0}\frac{\sin x\cos h+\sin h\cos x-\sin x}h\\ &= \sin x+x\cos x\lim_{h\to 0}\frac{\sin h}h+x\sin x\lim_{h\to 0}\frac{\cos h-1}h.\end{align}$$ It then remains to show that $\lim\limits_{h\to 0}\frac{\sin h}h=1$ and $\lim\limits_{h\to 0}\frac{\cos h-1}h=0,$ either as "special limits" or by some other means.
A: Depending on how familiar you are with the chain rule and derivative of logarithmic functions, you could differentiate $x \sin{x}$ using this alternate method that doesn't directly use the product rule,
$$\begin{align} y &= x \sin{x} \\
\ln(y) &= \ln( x \sin{x}) \\
\ln(y) &= \ln(x) + \ln(\sin{x}) \\
\frac{1}{y} \frac{dy}{dx} &= \frac{1}{x} + \frac{\cos{x}}{\sin{x}} \\
\frac{dy}{dx} &=  \left( \frac{1}{x} + \frac{\cos{x}}{\sin{x}} \right) \cdot y \\
\frac{dy}{dx} &=  \left( \frac{1}{x} + \frac{\cos{x}}{\sin{x}} \right) \cdot x \sin{x}\\
\frac{dy}{dx} &= \sin{x} + x \cos{x}. \\ \end{align}$$
A: A bit of intuition about the product rule:
Suppose that you have a rectangle whose height at time $t$ is $h(t)$ and whose width at time $t$ is $w(t)$. Then the area at time $t$ is $A(t)=h(t)w(t)$.  Now, as the time changes, how does the area change?

(Please, forgive my use of paint here.)
Say the white rectangle was from time $t$, and the larger rectangle at time $t+\Delta t$. We gain three new regions of area: the green one, the blue one, and the gray one.
The green area is $\Delta h\cdot w(t)$, where $\Delta h$ is the change in height from time $t$ to time $t+\Delta t$; the blue area is, similarly, $\Delta w\cdot h(t)$, and the gray area is $\Delta h\cdot\Delta w$. So, we have
$$
\Delta A=\Delta h\cdot w(t)+\Delta w\cdot h(t)+\Delta h\cdot\Delta w.
$$
Now, when $\Delta t$ is really small, we expect $\Delta h$ and $\Delta w$ to be really small as well; so, their product is tiny. Hence
$$
\Delta A\approx \Delta h\cdot w(t)+\Delta w\cdot h(t),
$$
or
$$
\frac{\Delta A}{\Delta t}\approx\frac{\Delta h}{\Delta t}\cdot w(t)+\frac{\Delta w}{\Delta t}\cdot h(t).
$$
Does this look at all like the product rule?  Letting $\Delta t\rightarrow0$, this approximation (properly formalized, of course) leads us to the formula
$$
\frac{d}{dt}\left[w(t)\cdot h(t)\right]=\frac{dA}{dt}=\frac{dh}{dt}\cdot w(t)+\frac{dw}{dt}\cdot h(t)
$$
A: You are probably expected to do this by the product rule, which says:
$(v \cdot w)'=vw'+v'w$
Setting $v=x$ and $w=\sin x$ (it doesn't matter which is which)
$v'=1$ and $w'=\cos x$ . Substituting into the formula:
$(v \cdot w)'=x \cos x+(1) \cdot \sin x$ 
$x \cos x+ \sin x$ 
which is your answer.
A: $x\sin(x)$ is the product of two functions. It's easier to see that there are two functions here if you rewrite components of the problem as $f(x)=x$ and $g(x)=\sin(x)$. If the two functions $f(x)$ and $g(x)$ are differentiable (the derivative exists), then their product is also differentiable.  
Now apply the product rule: ${(fg)}' = {f}'g+f{g}'$ to get $x\cos(x)+\sin(x)$.
