In physics the most common notations for differentiation are, $\frac{da}{dt} = \dot{a} = a'$. The notation $\frac{da}{dt}$ is due to Leibniz, the notation $\dot a$ is due to Newton, and the notation $a'$ is due to Lagrange. The dot notation, $\dot a$, almost always refers to a time derivative. The prime notation, $a'$, may refer to a derivative with respect to time or any other variable, typically made clear by the context, or by the authors explicit definition, for example

$$ \frac{da}{dx} = a'.$$

Typically, the quantity $\Delta a$ is not a derivative, but simply refers to the change in a itself,

$$ \Delta a = a_{final} - a_{initial}. $$

Knowing that, you can perhaps recall that $\Delta a$ is related to the definition of the derivative in the following way,

$$ \lim_{{t \to 0}} \frac{{\Delta a}}{{\Delta t}} = \frac{da}{dt} .$$

The fourth notation you've written, $\int_{i}^{f} a'$, is not a derivative and is not equal to the other expressions. It is also meaningless as an integral without the integration variable, and should look something like this,

$$ \int_{i}^{f} a' dt. $$

The fifth notation you've written, $\int_{i}^{f} a da$ is also not a derivative and is also not equal to the other expressions. It is a definite integral which can actually be solved. Perhaps if we wrote x instead of a this would be more clear,

$$\int_{i}^{f} x dx = \frac{x^2}{2} = \left. \frac{x^2}{2} \right|_{i}^{f} = \frac{(f-i)^2}{2}.$$

For a more comprehensive discussion of this topic there is a good article on Wikipedia. However, most of the article is beyond the scope of your question, so I would recommend you only refer to it to read about specific notations you've seen in class or in your textbooks that you're curious about.

Note: There is one additional notation for differentiation that is quite common in physics, however it refers to a related, but different concept called partial differentiation. For completeness it looks like this,

$$ \frac{\partial a}{\partial t} = \text{"partial derivative of } \textit{a} \text{ with respect to } \textit{t} \text{."}$$

In physics the most common notations for differentiation are, $\frac{da}{dt} = \dot{a} = a'$. The notation $\frac{da}{dt}$ is due to Leibniz, the notation $\dot a$ is due to Newton, and the notation $a'$ is due to Lagrange. The dot notation, $\dot a$, almost always refers to a time derivative. The prime notation, $a'$, may refer to a derivative with respect to time or any other variable, typically made clear by the context, or by the authors explicit definition, for example

$$ \frac{da}{dx} = a'.$$

Typically, the quantity $\Delta a$ is not a derivative, but simply refers to the change in a itself,

$$ \Delta a = a_{final} - a_{initial}. $$

Knowing that, you can perhaps recall that $\Delta a$ is related to the definition of the derivative in the following way,

$$ \lim_{{t \to 0}} \frac{{\Delta a}}{{\Delta t}} = \frac{da}{dt} .$$

The fourth notation you've written, $\int_{i}^{f} a'$, is not a derivative and is not equal to the other expressions. It is also meaningless as an integral without the integration variable, and should look something like this,

$$ \int_{i}^{f} a' dt. $$

The fifth notation you've written, $\int_{i}^{f} a da$ is also not a derivative and is also not equal to the other expressions. It is a definite integral which can actually be solved. Perhaps if we wrote x instead of a this would be more clear,

$$\int_{i}^{f} x dx = \frac{x^2}{2} = \left. \frac{x^2}{2} \right|_{i}^{f} = \frac{f^2-i^2}{2}.$$

For a more comprehensive discussion of this topic there is a good article on Wikipedia. However, most of the article is beyond the scope of your question, so I would recommend you only refer to it to read about specific notations you've seen in class or in your textbooks that you're curious about.

Note: There is one additional notation for differentiation that is quite common in physics, however it refers to a related, but different concept called partial differentiation. For completeness it looks like this,

$$ \frac{\partial a}{\partial t} = \text{"partial derivative of } \textit{a} \text{ with respect to } \textit{t} \text{."}$$

In physics the most common notations for differentiation are, $\frac{da}{dt} = \dot{a} = a'$. The notation $\frac{da}{dt}$ is due to Leibniz, the notation $\dot a$ is due to Newton, and the notation $a'$ is due to Lagrange. The dot notation, $\dot a$, almost always refers to a time derivative. The prime notation, $a'$, may refer to a derivative with respect to any other variable, typically made clear by the context, or by the authors explicit definition, for example

$$ \frac{da}{dx} = a'.$$

Typically, the quantity $\Delta a$ is not a derivative, but simply refers to the change in a itself,

$$ \Delta a = a_{final} - a_{initial}. $$

Knowing that, you can perhaps recall that $\Delta a$ is related to the definition of the derivative in the following way,

$$ \lim_{{t \to 0}} \frac{{\Delta a}}{{\Delta t}} = \frac{da}{dt} .$$

The fourth notation you've written, $\int_{i}^{f} a'$, is not a derivative and is not equal to the other expressions. It is also meaningless as an integral without the integration variable, and should look something like this,

$$ \int_{i}^{f} a' dt. $$

The fifth notation you've written, $\int_{i}^{f} a da$ is also not a derivative and is also not equal to the other expressions. It is a definite integral which can actually be solved. Perhaps if we wrote x instead of a this would be more clear,

$$\int_{i}^{f} x dx = \frac{x^2}{2} = \left. \frac{x^2}{2} \right|_{i}^{f} = \frac{(f-i)^2}{2}.$$

For a more comprehensive discussion of this topic there is a good article on Wikipedia. However, most of the article is beyond the scope of your question, so I would recommend you only refer to it to read about specific notations you've seen in class or in your textbooks that you're curious about.

Note: There is one additional notation for differentiation that is quite common in physics, however it refers to a related, but different concept called partial differentiation. For completeness it looks like this,

$$ \frac{\partial a}{\partial t} = \text{"partial derivative of } \textit{a} \text{ with respect to } \textit{t} \text{."}$$

In physics the most common notations for differentiation are, $\frac{da}{dt} = \dot{a} = a'$. The notation $\frac{da}{dt}$ is due to Leibniz, the notation $\dot a$ is due to Newton, and the notation $a'$ is due to Lagrange. The dot notation, $\dot a$, almost always refers to a time derivative. The prime notation, $a'$, may refer to a derivative with respect to time or any other variable, typically made clear by the context, or by the authors explicit definition, for example

$$ \frac{da}{dx} = a'.$$

Typically, the quantity $\Delta a$ is not a derivative, but simply refers to the change in a itself,

$$ \Delta a = a_{final} - a_{initial}. $$

Knowing that, you can perhaps recall that $\Delta a$ is related to the definition of the derivative in the following way,

$$ \lim_{{t \to 0}} \frac{{\Delta a}}{{\Delta t}} = \frac{da}{dt} .$$

The fourth notation you've written, $\int_{i}^{f} a'$, is not a derivative and is not equal to the other expressions. It is also meaningless as an integral without the integration variable, and should look something like this,

$$ \int_{i}^{f} a' dt. $$

The fifth notation you've written, $\int_{i}^{f} a da$ is also not a derivative and is also not equal to the other expressions. It is a definite integral which can actually be solved. Perhaps if we wrote x instead of a this would be more clear,

$$\int_{i}^{f} x dx = \frac{x^2}{2} = \left. \frac{x^2}{2} \right|_{i}^{f} = \frac{(f-i)^2}{2}.$$

For a more comprehensive discussion of this topic there is a good article on Wikipedia. However, most of the article is beyond the scope of your question, so I would recommend you only refer to it to read about specific notations you've seen in class or in your textbooks that you're curious about.

Note: There is one additional notation for differentiation that is quite common in physics, however it refers to a related, but different concept called partial differentiation. For completeness it looks like this,

$$ \frac{\partial a}{\partial t} = \text{"partial derivative of } \textit{a} \text{ with respect to } \textit{t} \text{."}$$