Math, Calculus, How can we square d sign in derivative of second order? I can say that I'm pretty good at math. I always liked math and understood it, BUT one thing is almost killing me. As I know sign d in derivative is equal to greek sign delta which means the difference. So if it is only sign that shows difference, how can we use it as an operator and square it in derivatives of second order? I mean d^2y/dx^2. My teacher said just accept it and go on but how can I accept something that I don't understand? This thing irritates me the most.
 A: The notation you are calling $\frac{d}{dx}$ can be interpreted in a slightly more general way than just as a thing which eats functions and returns their derivatives. Since $\frac{d}{dx}$ satisfies $\frac{d}{dx}(f+g) = \frac{d}{dx}f + \frac{d}{dx}g$ and $\frac{d}{dx}(cf) = c \frac{d}{dx}f,$ it gives a linear function on the set of all differentiable functions $\mathbb{R} \to \mathbb{R}$.
Now you may know that when you have a linear map on a space, it has a representation as a matrix. $\frac{d}{dx}$ does not have such a representation, since the space of all differentiable functions is not finite dimensional, and we can't write down a matrix. However, when you have two linear maps, you may know that composing them is the same as multiplying the matrices. Let's say for example that $T: V \to V$ was a linear map. Since it's codomain is also $V$, we can apply $T$ twice, and in this case we would write $T \cdot T = T^2$ for that composition.
If you have a function which is twice differentiable, then you can apply $\frac{d}{dx}$ twice to that space, so even though we can't write it down as a matrix, the idea for the notation might be thought of as the same.
I should clarify that this notation is kind of old, and probably this analogy I have in my head was not around when the notation was invented, but this is how I think of the notation - we just square the operator because it's like composing two linear maps.
A: $\frac{d^2 y}{dx^2}$ is the notation we use to signify $\frac{d}{dx}(\frac{dy}{dx})$. This makes is more concise to write down, especially when doing more repeated derivatives: for example $\frac{d^5y}{dy^5}$ is shorter and simpler to read than $\frac{d}{dx}(\frac{d}{dx}(\frac{d}{dx}(\frac{d}{dx}(\frac{dy}{dx}))))$
There are ways of interpreting this 'difference of a difference' as a 'difference squared', but we aren't usually thinking of $\frac{d^2y}{dx^2}$ as a square like $f(x)^2$ or $3^2$.
