Answer to Question #91370 in Calculus for Ismail

Question

First principle of differentiation

Answer 1

We know that the straight line has a constant gradient or we can also say that the rate of change of y with respect to x is a constant. One can define the instantaneous rate of change of y with respect to x. This is equivalent to the following:

$$\frac{{dy}}{{dx}} = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{\Delta y}}{{\Delta x}}\\ \frac{{dy}}{{dx}} = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{f\left( {x + \Delta x} \right) - f\left( x \right)}}{{\Delta x}} .$$

It is also known as the delta method.