# Derivatives

By definition the derivate is the slope of the tangent line that passing through a unique point of a function, in others cases is define as the change rate of this.

How know calulate the derivate of a function?, as before we said, is the slope of the tangent line to function. How calculate the slope of a function, considering 2 points?.

By definiton the slope to line is:
Considering 2 points (x,y) and (x0,y0) in a plain.

$\bg_white \small \color{black} \displaystyle{(y-y_0) \over ({x-x_0})}$

Now, when will we say that this slope is in a only point. we can observe in the formula that is wrote before, that if we use an only point, the formula denominator is indeterminate becouse (x-x) is equal zero.

Well, from this is we say that a derivatives (slope of the tangent line) is a limite of function in 2 distinct points when distance between these points becomes very small, and is defined with next formula.

$\bg_white \small \color{black} \lim_{x0 \to x} \displaystyle{f(x)-f(x0) \over x-x0}$

For example
If we need calculate the function derivate of x², using the formula before. is solved to the next way.
$\bg_white \small \color{black} f(x) = x^2$

therefore, limits apply, we will say
$\bg_white \small \color{black} f'(x) = \lim_{x0 \to x} \displaystyle{f(x)-f(x0) \over x-x0}$

and solved the limits
$\bg_white \small \color{black} f'(x) = \lim_{x0 \to x} \displaystyle{x^2-x0^2 \over x-x0}$