PjProblemStrings Sequencing And The Derivative.
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The derivatives and anti-derivatives of functions are very important analytical tools. They form the branch of mathematics generally known as calculus. Calculus has been and continue to be an indispensable tool in the analysis of existential problems. The great Isaac Newton (1642-1727 A.D.) formulated the calculus. He called his formulation the method of Fluxions.

(a) What work does the derivative do?
(b) Use PjProblemStrings Sequencing to broadly describe this work of the derivative.
(c) What is the geometric interpretation of the derivative of a single variable function and a multi-variable function.
(d) Is the geometric interpretation consistent with PjProblemStrings Sequencing?
(e) What are some of the applications of the derivative?

The strings: S₇P₃A_3k (force. k =1, 2), S₇P₄A_4k (motion. k =1, 2, 3,4), S₇P₅A_5k (change. k =1, 2). Where k is not a multiplicand.

The math:
All Pj Problems are at play. However, Pj Problems of Interest are of types force, motion and change.

(a)The derivative as an analytic tool, measures relational dynamism. In other words, a variable change (change in a dependent variable) that is caused by another variable change (change in an independent variable). The process of deriving a derivative is called differentiation.

(b)Space dynamism is a fundamental existential phenomenon. It is caused by forces. A force is either a pull or a push. The PjProblemStrings representing these forces are:
- S₇P₃A₃₁ (a multi-matter dynamic space in which the dynamism of interest is caused by pulls)
- S₇P₃A₃₂ (a multi-matter dynamic space in which the dynamism of interest is caused by pushes).
The dynamism of a space may be caused by pushes and pulls acting concurrently.

PjProblemStrings sequencing describes dynamism as motion sandwiched between two forces. The PjProblemStrings representation is:
- S_iP₃A_3kS_iP₄A_4kS_iP₃A_3k. Where S_iP₄A_4k represent motion and k is not a multiplicand.

The first force is the active force and the second is resistive force. The difference between the active force and the resistive force is the resultant force (driving force) causing the motion. In some scenarios, the active force is deliberately reduced in order to reduce or stop the motion (e.g. a runner reduces speed after crossing the finishing line). When the resultant force is zero, there is static equilibrium. When the resultant force is constant, there is dynamic equilibrium. The PjproblemStrings of change are implicit in the dynamism.

(c)The derivative is defined as the slope of a function at a point.



In figure 1a, there is no change in the dependent variable relative to the change in the independent variable. So, the slope is zero and by definition, the derivative is zero.
In figure 1b, the change in the dependent variable relative to the change in the independent variable is constant. So, the slope is a constant and so is the derivative.
In figure 1c, the dependent variable changes at every point relative to the change in the independent variable. So the slope varies at every point of the curve. Therefore, the slope at a point is the slope of a line tangent to the point. The slope of the tangent line is the slope of the secant line as the change in the independent variable tends to zero.
Figure 1d is a geometric interpretation of the partial derivative at a point of a multi-variable function. The derivative is a partial derivative because there are other variables at play. Only one independent variable is allowed to change at a time, relative to the change in the dependent variable. See geometric interpretation of the partial derivative.

(d)Yes. In all of the cases in figure 1, one can imagine the PjProblemStrings Sequencing necessary to realize the desired slope. For example, the derivative (slope) at point P is realized by the movement of point Q towards point P as the secant PQ tends towards the tangent line at P.

(e)The derivative is used to measure rate of change. For example, instantaneous velocity and acceleration. In economics it is used to measure marginal cost, marginal profit and many other optimization problems in various disciplines.

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