If given a function $f$ that I am supposed to demonstrate is derivable, is it enough for me to try to compute its derivative f' and if the result is continuous, can I then state that the original function $f$ is derivable?
How to demonstrate derivability?
0
$\begingroup$
derivatives
-
1I guess you mean differentiable. If the derivative *exists* at some point, then the function is differentiable there. If the derivative *exists and is continuous*, the function is said to be continuously differentiable. – 2012-01-22
1 Answers
2
In most cases you would distinguish two situations:
- A region where $f$ is clearly derivable because it is a composition of elementary functions known to be derivable (polynmials, exponentials, trigonometric functions,...)
- Some points where it is not clear what happens, and you have to show that the derivative exists using the definition.
Let me give an example: $f(x)=x^2\sin\dfrac{1}{x}$ if $x\ne0$, $f(0)=0$. $f$ is derivable in $\mathbb{R}\setminus\{0\}$ because it is a composition and product of derivable functions. What hapens at $x=0$? Using the definition of derivative, you can show that f'(0)=0, s0 that $f$ is derivable in $\mathbb{R}$. Observe hat f' is not continuous at $x=0$ (the limit as $x\to0$ of f'(x) does not exist.) Continuity of f' is not a necessary condition for $f$ to be derivable. Functions with a continuous derivative are called $C^1$ functions.