Instantaneous Rate of Change

$Slope=\frac{dy}{dx}$ $evaluatedat\left(x_1,y_1\right)$

And this is the formula for the instantaneous rate of change of y with respect to x at that point.

$\frac{dx}{dy}=\frac{1}{\frac{dy}{dx}}$

Recall that the average rate of change is taken over a big interval and is given by $\frac{\Delta y}{\Delta x}$ for a function$y=f\left(x\right)$.

If we bring points A and B closer and closer to each other, they will eventually become a single point. This is the reason we use the limiting condition $\underset{\Delta x\to 0}{\lim }$ and $\underset{\Delta y\to 0}{\lim }$ to avoid both points merging together.

We can name the coordinates at A as $\left(x,f\left(x\right)\right)$ and the let the coordinates of point B as $(x+a,f\left(x+a\right))$ since the values will vary very slightly as they are extremely close to each other. Hence we have $x_1=x$, $x_2=x+a$ and $y_1=f\left(x\right)$, $y_2=f\left(x+a\right)$.

$\frac{\Delta y}{\Delta x}=\frac{f\left(x+a\right)-f\left(x\right)}{x+a-x}$$=\frac{f\left(x+a\right)-f\left(x\right)}{a}$

$\underset{a\to 0}{\lim }\frac{\Delta y}{\Delta x}$$=\underset{a\to 0}{\lim }\frac{f\left(x+a\right)-f\left(x\right)}{a}$

$\frac{dy}{dx}=\underset{a\to 0}{\lim }\frac{f\left(x+a\right)-f\left(x\right)}{a}$