Definition: the tangent line is the slope of a very specific point of the function.
to find the slop for point $P(a, f(a))$, we use a very close but not the same point, $Q(x, f(x))$ in this formula
$$m_{PQ} = \lim\limits_{x \to a} \frac{f(x) - f(a)}{x - a}$$
then the tangent, $t$, is the line through $P$ with slope $m$. from there you can use point slope form, $y - y_1 = m(x - x_1)$, to find the equation of the line. The following equation can also work to find the slope (this is achieved by making h = a + x):
$$m_{PQ} = \lim\limits_{h \to 0} \frac{f(a + h) - f(a)}{h}$$
for what its worth, this equation can also be used to find velocity
because the above function is so incredibly common, we give it a special name, a derivative
$$f’(a) = \lim\limits_{h \to 0} \frac{f(a + h) - f(a)}{h}$$
if we have to x values, we can define the change from one to te other as $ \Delta = x_2 - x_1$. then change in y values could then be $ \Delta y = y_1 - y_2$. if we wanted to track how often the y changed per x change, the rate of change, we could write it as:
$$ \frac{\Delta y}{ \Delta x} = \frac{f(x_3) - f(x_1)}{x_2 - x_1}$$
if we wanted to use this for to find the rate of change for a particular point, we make $x_2$ approach 0 like this:
$$ \lim\limits_{\Delta x \to 0} \frac{\Delta y}{ \Delta x} = \lim\limits_{x_2 \to x_1} \frac{f(x_3) - f(x_1)}{x_2 - x_1}$$
This is the Instantaneous Rate of Change. The derivative f’(a) is the instantaneous rate of change of y = f(x) with respect to x when x − a.