Question 943835
The difference quotient is defined by *[tex \large \frac{\Delta g(x)}{\Delta x} = \frac{g(x + \Delta x) - g(x)}{\Delta x}]


Substituting we get


*[tex \large \frac{\Delta g(x)}{\Delta x} = \frac{(x + \Delta x)^2 - x^2}{\Delta x} = \frac{2x \Delta x + \Delta x^2}{\Delta x} = 2x + \Delta x]


Note that as *[tex \large \delta x \to 0], we obtain the expression for the derivative of x^2 with respect to x, which is 2x.