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I know that a function is differentiable if the limit exists as $\Delta x \to 0$ of a certain limit. But how can one know this beforehand?

I mean, we usually just differentiate using rules that we have derived using the limit definition, but then how we do know that the function we apply them to actually did pass the "limit test" and is differentiable?

Is there some way to just look at a function and know whether it's differentiable or not? And not just for single variable, but for multivariables as well?

Jack Dan
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    In a formal constitution of calculus, you'd have to prove that things like polynomials and certain power series are differentiable. And all rational function (using $+,-,\cdot,/$) of differentiable functions are aswell (except for their poles), etc. From their on, you can immediately judge the differentiability of a given function. – GDumphart Oct 12 '14 at 10:43

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The rules to which you refer not only tell you what the derivative is but, also, that it is differentiable in the first place. For example, a complicated looking function like $$f(x) = e^{-x^2} \cos ^2(x) \left(x^3-4 x+\cos (\pi x+1)\right)$$ is certainly differentiable at every point because it is a combination involving sums, products and composition of power functions, the cosine and the exponential function, which are known to be differentiable. On the other hand, $g(x)=\sin(1/x)$ and $h(x)=x^{2/3}$ involve a quotient and a root, which both cause problems at the origin.

Mark McClure
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There is no as such general rule which applies for every function but there are some rules of thumb which you may use.

1.Every polynomial function is differentiable at all points. eg. - $x^4 - 5x^2$.

2.If you add or subtract two differentiable functions(i.e. which are differentiable at all points), the result is also a function which is differentiable at all points. eg - $\sin x + \cos x$. Since both $\sin x$ and $\cos x$ are differentiable at all points so is their sum.

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