Lesson 3 Derivatives

Kinematics is a branch of science that deals with motion in the abstract. In particular, particle kinematics deals with position, distance, velocity, speed, and acceleration. In order to understand these components of motion and their presence in the study of physics, one must explore derivatives. A derivative is said to be the "grammar of Differential Calculus" and details three major items. First, a derivative provides the slope of a line tangent to a curve at any point on the curve, Second, a derivative provides the velocity of an object at any time, and Third, a derivative provides the rate of change of a variable with respect to another variable. Slope, a major factor of Differentiation, is also a derivative:

Positive and Negative Slopes

Slope of a chord = Change in elevation                    (On a curve)

                        Change in horizontal distance

Slope of tangent line = derivative

Therefore, slope is the derivative of elevation with respect to horizontal distance.
Slope of tangent = dy
                               dx

There are also a number of rules that govern the ways in which a function is derived:

The Sum Rule

d (mx+b) = m

dx

d (y+z) = dy + dz

dx            dx    dx

The Product Rule

A= l x w

(Delta)A= (Delta)l x w

(Delta)A = l x (Delta)w

d (yz) = ydz + zdy

dx            dx      dx

d (xx) = x d x + x d x

dx              dx        dx

d x^2 = 2x d x

dx              dx

d (mx + b) = m

dx

d (1x + b) = 1

dx

d x = 1

dx

d x^2 = 2x

dx

d x^n = nx^n-1

dx

The Chain Rule

dy = dy dx

dt     dx dt

All of these rules contribute to the ease of mathematics when exploring complex theories. For example, Albert Einstein's Theory of Relativity is the most complex aspect of physics, and Einstein stated that he longed to use the simplicity of mathematics to support his theory. However, his theory was so elaborate that he could not rely on mathematics eniterly, but rather, use it as a tool because mathematics is limited in some scenarios. For instance, it is impossible to take the derivative of an absolute value function when x = 0. From Kinematics to calculus, mathematics and Differentiation are undeniably important.

*(Watch the following video for some in-depth studies of derivatives)*

Throughout this video you will view the concept of a derivative through the study of slope, elevation, and horizontal distance. The "Derivative Machine" will display the derivatives of trigonometric functions. The video concludes with the relationship between distance, velocity, and acceleration. The three graphs portray relationships between the graph of a function and its derivative. In fact, the derivative of a function is negative (below x-axis) when the function decreases and the derivative is positive (above x-axis) when the function increases.