Question

I'd like to find a resource that explains and gives the proof of where the formula for a Derivative of a Function comes from. I don't necessarily expect to understand it all, but if anyone knows of one that is mildly friendly to the average joe, please, let me know.

Answer 1

I think it's in the first video lecture and it's explained pretty well.

Basically, the formula for a derivative takes two points of a function f(x) and gives you the slope of the line that connects the two points. The slope is the same as in basic algebra: rise over run. In calculus it is called the change in f(x) divided by the change in x. You take two values of x, say x1 and x2, and find f(x1) and f(x2). The slope of the line connecting those two points is:\[(f(x _{1})-f(x_{2}))\div(x_{1}-x_{2})\]

As you let the change in x get closer to zero - in other words as the two points become closer and closer to each other - the line connecting the two points becomes a tangent line to the function and the ratio of the change in f(x) over the change in x gives you the slope of that tangent line. The derivative then is a function which gives you the slope of the tangent line at any value of x (provided the function is differentiable).

Check out the animation on this page for a visual reference: http://www.sosmath.com/calculus/diff/der00/der00.html