Show that the functions are differentiable I have two functions $$f(x)=x\sin^2\frac{x}{x^2+1}$$
$$g(x)=\sqrt{\sin\frac{x^2+1}{x^2+2}}$$
Now I have to show the functions are differentiable, and if they are I can then determine the derivative. According to the definition I shall show that: $$\lim\limits_{h\to 0}{\frac{f(x_0+h)-f(x_0)}{h}} $$ exists.
But if I put the $f(x)$ or $g(x)$ inside the limit it gets like really nasty. I don't even think I was able to compute such a limit.
Can you guys give me any hints? How could I show these functions are differentiable without computing the derivatives first?
 A: You would normally use some basic facts beyond the mere definition of a derivative (theorems to be proven).
Constants are differentiable with derivative 0.
The identity mapping $x$ is differentiable with derivative 1.
Constant multiples of differentiable functions are differentiable, and the derivative is multiplied by the constant.
The sum of two differentiable functions is differentiable, and their derivative is the sum of the individual derivatives.
The product of two differentiable functions $f$ and $g$ is differentiable, and their derivative is $fg'+f'g.$
The composition of differentiable mappings is differentiable and that the derivative of the composition is the product of the individual derivatives (a.k.a. the Chain Rule)
The sine function is differentiable and its derivative is the cosine.
The function that maps $x$ to its reciprocal $1/x$ is differentiable everywhere except at $x=0$ and its derivative is $-1/x^2.$
The square root function is differentiable on the strictly positive real numbers and its derivative is $\frac1{2\sqrt x}.$
A: If you're being asked to show if the functions are differentiable in a certain interval, it suffices to proof that the functions are smooth for said interval. Now, if you're being asked to show that Limit exists, you'll have to bite the bullet.
A: Hint: If you've covered the differentiation rules up through the chain rule, then the question on $f(x)$ is a slam-dunk. The second question on $g(x)$ adds a little bit of head-scratching, but if you can identify the range of $(x^2+1)/(x^2+2)$ and recall where exactly $\sqrt x\,$ is differentiable, then it's full steam ahead.
