I know how to derive, I know how to integrate. I know what to do when I see $\frac{d}{dx}$ and such but what does it really mean? I know it means something like derive in terms of $x$, but whats the difference between $\frac{dy}{dx}$ and $\frac{d}{dx}$?
If someone could give me an explanation in layman's terms that would be very helpful as this has always perplexed me.
Basically, what's that $d$ mean?
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$\begingroup$If you have a function $f$ with the independent variable $x$, then $$ \frac{d}{dx} f(x) $$ means the derivative of $f$ with respect to $x$. We also sometimes write this as $f'(x)$. Now if you have a function like $$ f(x) = ax, $$ then the derivative is $$ f'(x) = a. $$ This is clear because in writing $f(x)$ we have indicated that the function $f$ is a function of the variable $x$. If I instead had told you that $$ y = ax $$ and I just asked you to find the derivative, what would you do? You would probably, again, just say that the derivative is $a$. But in this situation it actually isn't clear what is a variable and what is a constant. And therefore we can write $$ \frac{dy}{dx} \quad\text{or}\quad \frac{d}{dx}y $$ to indicate that we are considering $y$ as a function of the variable $x$ and we are considering $a$ as a constant (In the case of multivariable functions we really should be taking about partial derivatives in this case). Now you could also write $$ \frac{dy}{da} $$ and in this case you are saying that $a$ is a variable. So the $\frac{d}{dx}$ notation is very helpful when you have expressions where there are several letters.
So what is $\frac{d}{dx}$? You can consider this as an operator that takes as "input" a (differentiable) function and "outputs" a function.
$\endgroup$ 1 $\begingroup$$\dfrac{dy}{dx} = \dfrac{d(y)}{dx} = \dfrac{d}{dx}(y)$.
$\dfrac{d}{dx}$ is the differential operator. It tells you what operation (differentiation) you are doing and with respect to what variable. $\dfrac{dy}{dx}$ is the actual derivative of a function $y$ with respect to $x$. The operator can be applied to any function, for example $\dfrac{d}{dx}(x^2) = 2x$. If we wrote the same thing with $\dfrac{dy}{dx}$, then we get $\dfrac{dy}{dx}(x^2) = x^2\dfrac{dy}{dx} = x^2y'$. Since $\dfrac{dy}{dx}$ is already differentiating the function $y$, it does not do anything to $x^2$. The former example shows the differential operator being applied to a function. The latter shows the derivative of $y$ being multiplied by a function. The $x^2y'$ expression is often used in differential equations as shorthand to replace the longer to write $\dfrac{dy}{dx}$.
$\endgroup$ $\begingroup$$\dfrac{d}{dx}$ is what analysts would call an operator, meaning that you give it an element of a certain vector space and it gives you another element in that vector space. So then what is a vector space? Well a vector space is any collection of objects that satisfy certain axioms (like you can add two of of these objects together to get another object in the collection, you can multiply an object with a number to get another object in the collection, every object has a negative, etc). Vector spaces are in some way a generalization of the real numbers.
With that out of the way, how does this relate to $\dfrac{d}{dx}$? Well, you can look at $\dfrac{d}{dx}f$ for some function $f$. This is nothing more than the derivative of $f$. Particularly, there is a very nice vector space of functions called the "smooth" functions. These functions are infinitely differentiable and every derivative is continuous (though this is a bit redundant to say). Examples of such functions are polynomials (but these are not all of the smooth functions). Another would be $\arctan(x)$. If you differentiate a smooth function, you get another smooth function! So then if we denote the smooth functions by the symbol $C^{\infty}(\mathbb{R})$, we have that $\dfrac{d}{dx}:C^{\infty}(\mathbb{R})\rightarrow C^{\infty}(\mathbb{R})$.
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