Unit V Big Question

Where does the formula from the difference quotient come form? 
1. A derivative is the slope of a tangent line to a graph. A tangent line is a line that touches the graph at one point. A function can have different derivatives at different values of x. To try and find the slope of the tangent line we first have to find the slope of a secant line similar to the tangent line. A secant line is a line that touches the graph at 2 points. We find the slope of the secant line using the slope formula using the 2 points on the secant lines. This is shown in the picture below. When we plug in our points we get m= f(x+h)-f(x)/ x+h-x. So when we simplify the denominator we are left with 
m=f(x+h)-f(x)/ h. So this is what we know as the difference quotient and is also the slope of the secant line. But remember we want to find the slope of the tangent line. So what we will notice is that as you make h(which is the change in x or the difference between the two x points) smaller, the secant line begins to resemble the tangent line better. So we want h(or delta x) to be as small as possible. So we use a limit as h approaches 0. We cannot just say that h=0 because then we will have a 0 in the denominator, so this is why we use a limit. By using a limit of 0, we are now finding the slope of the tangent line. 

Image: http://en.wikipedia.org/wiki/Numerical_differentiation