1. The History of Calculus:
Newton vs Leibniz
Isaac Newton
Gottfried Leibniz
The calculus controversy was an argument between seventeenth-century mathematicians Isaac Newton and Gottfried Leibniz over who had first invented calculus. Newton claimed to have begun working on a form of the calculus (which he called "the method of fluxions and fluents") in 1666, but did not publish it except as a minor annotation in the back of one of his publications decades later. Gottfried Leibniz began working on his variant of the calculus in 1674, and in 1684 published his first paper employing it. L'Hopital published a text on Leibniz's calculus in Meanwhile, Newton did not explain his calculus in print until 1693 (in part) and 1704 (in full). While visiting London in 1676, Leibniz was shown at least one unpublished manuscript by Newton, raising the question as to whether or not Leibniz's work was actually based upon Newton's idea. It is a question that had been the cause of a major intellectual controversy over who first discovered calculus, one that began simmering in 1699 and broke out in full force in 1711.
— from Wikipedia, the free encyclopedia

2. The History of Calculus:
Newton vs Leibniz
Developed Differential Calculus to help solve problems in Physics
By discovering the rate of change of the area function, he realized area could be computed using antiderivatives
Did not believe in using infinitesimals, stating only the ratio dy/dx made any sense, and that dx by itself is meaningless
Developed Integral Calculus to help solve problems in Geometry (also believed philosophical truths could be reduced to calculation)
Discovered the area under a curve was related to the slope of a tangent line
Had no problem with infinitesimals, inventing the notation dx for differentials (also invented the  symbol)

3. Dynamical Systems Static Geometry
A derivative dy/dx represents motion or relative changes in variables x and y but measured at a single point: “not before, and not after, but exactly at that point”
A curve in the plane represents the trajectory of a moving body, and as it moves along the curve the derivatives tell in which direction the body moves next
As the body moves along the curve, it sweeps through little increments of area, and if these are summed to define an area function, then when the curve is defined by some function y=f(x) we can calculate the area function because the derivative of the area function is given by f
Given a curve y = f(x), which is a static set of points {(x,y)} in the plane, the derivative dy/dx gives the slope of the line tangent to the curve at a point
The derivatives of f define the shape of the curve, and if we already know what the shape of the curve is, we can work backwards and figure out what f must be
Taking this working backwards one step more, the area under a curve defined by f is given by finding an antiderivative of f
These two ways of describing calculus are equivalent, and both lead to the Fundamental Theorem of Calculus.

4. Derivatives and Differentiability
Given a curve y = f(x) with f continuous for a < x < b, the slope of the secant line joining the points (a,f(a)) and (b,f(b)) is given by x y
y = f(x)
tangent
a+Dx
secant a b
f(a)
f(b)
This also defines the difference quotient
Taking the limit as a  b (or, equivalently, as Dx  0), gives the slope of the tangent line at the point (a,f(a)):
If mtan exists and is equal to the limiting value of msec then f is said to be differentiable and dy/dx is called the derivative of f.

5. Equation for the Tangent Line
Given a curve y = f(x) with f continuously differentiable at x = x0, with y0 = f(x0), the slope of the tangent line is given by the derivative f (x0). x y
y = f(x)
tangent x0
y = L(x)
If (x, y) is any other point on the tangent line, then the slope of the line joining (x, y) and (x0, y0) must be equal to f(x0):
If f is continuously differentiable at x0, the function L(x) = f(x0) + f (x0)(x – x0) is called the linearization of f at x0.