# What’s differentiation?

In my first lesson I talked about integration: in a nutshell, it’s the mathematical operation used to calculate the area under a curve of function f(x). The integral of a function is also called antiderivative. But wait, if this is “anti”, what is the derivative of a function? Well, simply said, it is the inverse of an integral, which means that, having the function that describes the area under a curve, differentiating (calculating the derivative of) this function gives the function of the curve. So, differentiation is the action of computing the derivative of a function, the same way integration means “calculating the integral of”. This means that the integral (or antiderivative) of the derivative of a function is that function and the other way around.

The symbol of derivative is

$\frac{d}{dx}$

and the derivative of a function f(x) is written as

$\frac{d}{dx}f(x)$

which is read “d dx of f of x” or “(first) derivative of f of x”,

or

$\frac{\mathrm{d}&space;y}{\mathrm{d}&space;x}$

“d y d x”

or even

$f'(x)$

which is read as “f prime of x”.

As I said before:

$\int\frac{d}{dx}f(x)dx=f(x)$

and

$\frac{d}{dx}\int&space;f(x)dx=f(x)$

The limit definition of derivative is

$\frac{\mathrm{d}&space;}{\mathrm{d}&space;x}f(x)=\lim_{h&space;\to&space;0}\frac{f(x+h)-f(x)}{h}$

As an example, let’s take a parabola of function f(x) = x² :

In the next two posts I will teach you how to calculate the derivative and the integral of a function, but for now I will just assume you already know to make the concept more clear.

The area under this parabola is calculated using the indefinite integral of x² . Such integral tells you the formula of the area, the same way h times b is the formula to calculate the area of a rectangle. To actually know the numerical value of the area you need to calculate the definite integral of x². I covered these two types of integral in my post about integration.

$F(x)&space;=&space;\int&space;x^{2}dx=\frac{x^{3}}{3}+C$

F(x) is the antiderivative (integral). So x³/3 is the formula of the area under this parabola. If I wanted to know what the function of the parabola is, I would have to differentiate (calculate the derivative of) the formula for the area, here x³/3 :

$f(x)=\frac{d}{dx}\frac{x^{3}}{3}=x^{2}$

This can be verified since we said before that

$\frac{d}{dx}\int&space;f(x)dx=f(x)$

Now you should know what we mean with differention and be ready to learn to actually calculate the derivative of a function. Stay tuned for more!

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