Is $f'(a)$ Not same as Finding Derivative of $f(x)$ and Substituting $x=a$ I got this doubt regarding the differentiability of function $f(x)$ at $x=0$ where
$$f(x)= \begin{cases}
            x^2 \sin\left(\frac 1 x\right) & x \ne 0\\
            0 & x=0
         \end{cases}
$$
We know that $f(x)$ is Differentiable at $x=0$ and the Slope of tangent at $x=0$ is Zero Since RHD = LHD
But i am getting confused of the notation many books use for RHD = LHD = $f'(a)$
Because $f'(a)$ is value of the function's Derivative at $x=a$ in this case we have
$$f'(x)=-\cos \left(\frac{1}{x}\right)+2x \sin\left(\frac{1}{x}\right)$$ So $f'(0)$ Does not exist. But since $f(x)$ is differentiable at $x=0$ the notation says $f'(0)=0$
So i need clarification in this ?
 A: When you found the expression for $f'(x)$, the expression you used for $f(x)$ isn't valid for $x = 0$. Therefore there is no reason to expect the expression you find for $f'(x)$ to be valid for $x = 0$ either.
For $x = 0$, it's better to turn to the definition:
$$
f'(0) = \lim_{h\to 0}\,\frac{f(h) - f(0)}{h}\\
= \lim_{h\to 0}\,\frac{h^2\sin(1/h)}{h}\\
= \lim_{h\to 0}\,h\sin(1/h) = 0
$$
Here is a graph of the function $f$, close to the origin:

You can see how each wave oscillates wildly (I've stretched the $y$-axis some, so it's exaggerated), but as a whole the graph flattens out, as it's squeezed between the two parabolas $y = x^2$ and $y = -x^2$. This is the reason it can be differentiable at $0$ even though the expression for $f'(x)$ doesn't make sense or even has a limit as $x\to 0$.
As a second way to look at this, if I hadn't stretched out the $y$-axis in the graph above, all we would have seen would be a somewhat thick line along the $x$-axis. Which means that close to the origin, the line $y = 0$ is actually a good approximation to $f(x)$, close to $x = 0$. The existence of a line which is a good approximation, under a certain strictly defined sense of "good approximation" (basically, no matter how much I stretch the $y$-axis to make the approximation look bad, just zooming in will eventually make it look good again), is the definition of differentiable, and a notion which is readily generalized to higher dimensions.
This function is the standard example illustrating the difference between a a function being differentiable and a function being continuously differentiable.
