Introduction to derivatives and integrals question I have recently started studying calculus and I constantly find my self baffled by its notations. Specifically can somebody explain the difference between:
$$\int_{a}^{b} x^2 dx$$ $$\int_{}x^2 $$ $$\int_{a}^{b} x^2$$
and $$\frac{d}{dt}x^2$$ $$\frac{dx}{dt}x^2$$
 A: Disclaimer: These are my interpretations of the notation, but other people may have other ways of writing things.
The first step is noting that
$$\int_a^bf(x)\ dx=\text{Area}$$
while
$$\int f(x)\ dx=\text{anti-derivative}$$
These are two separate concepts, bound together by the fundamental theorem of calculus:
$$\int_a^bf(x)\ dx=F(b)-F(a),\text{ where }F(x)=\int f(x)\ dx$$
Anti-derivatives are defined as follows:
$$\frac d{dx}F(x)=f(x)\implies F(x)=\int f(x)\ dx$$
It is, lastly, in my greatest opinion that $\int f(x)$ is unclear notation.  A lack of that final $dx$ will cause you great misfortune, for example, what does the following mean?
$$\iint\frac xy\sin(z)=?$$
Does it mean to integrate with respect to $x,y$ or $z$?  And in which order?  Clearly, this expression makes no sense, but,
$$\iint\frac xy\sin(z)\ dx\ dz$$
Makes perfect sense.

Derivatives, on the other hand, have their own tricky notations:
The following means to take the derivative with respect to $t$ for everything after the $\frac d{dt}$:
$$\frac d{dt}x^2$$
But if there is something in the numerator, like
$$\frac{dx}{dt}x^2$$
It means to take the derivative of the numerator with respect to $t$, then multiply it by whatever comes after.
A: In all, I saw $5$ mathematical expressions:


*

*The first one is a definite integral of $x^2$ from $a$ to $b$.

*No meaning as it contains no differential. If it contained a differential, it would be an indefinite integral or an antiderivative.

*No meaning due to above reason.

*Derivative of $x^2$ with respect to $t$

*Product of $x^2$ with the derivative of $x$ with respect to $t$
These are what they actually are.
A: The users above have sufficiently explained the difference between the integral notations, so I won't go over it again, but you asked for an explanation of the difference between $\frac {dx}{dt}  x^2$ and $\frac d{dt} x^2$.
Well let's say that $x = f(t) = 2t^2$. Then the evaluation of the first expression would be to differentiate $x$ with respect to $t$ and multiply it by $x^2$: $\frac{dx}{dt} \times x^2  =4tx^2$.
The second one is a bit harder to evaluate. In this one, we are differentiating $x^2$ with respect to $t$. However, $x$ is a function of $t$ and does not contain any terms of $t$ in it, so how do we differentiate it? Well, in this case, we can rewrite $x^2$ using the function of $t$, but in other cases, we can't just do that. Instead, we use a method called implicit differentiation.
