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Calculus: Derivatives (Part 3 of x)

Preface
This series is aimed at providing tools for an electrical engineer to analyze data and solve problems in design. The focus is on applying calculus to equations or physical systems.

Introduction
This article will cover derivatives.

There are many calculus references, the one I like to use is Calculus by Larson, Edwards and Hostetler.

This also assumes you are familiar with Python or can stumble your way through it.

Concepts
None

Definition of a Derivative
For a function of one variable: the derivative of f(x) at x is given by
f'(x) = [f(x+dx)-f(x)]/[dx] = m, the slope of the line
provided the limit exists.

For a function of more than one variable, partial: the partial derivative of f(x,y,z) at x is given by
∂/∂x = fx(x,y,z) = [f(x+dx,y,z)-f(x,y,z)]/[dx]
provided the limit exists.

Differentiability implies continuity. But continuity does not imply differentiability.

Properties of Derivatives

General Form u dx
Constant c dx = 0
Constant Multiple cu dx = cu'
Chain f(u) dx = f'(u)u'
General Power un dx = nun-1u'
Sum/Difference [u +/- v] dx = u' +/- v'
Product uv dx = uv' + vu'
Quotient (u/v) dx = (vu' - uv')/v2

Where u and v are a functions, c and n are constants.

Derivative Table
Here.

Derivatives of Vector Functions
If r is vector-valued function r(t) = f(t)i + g(t)j + h(t)k, then

r'(t) = f'(t)i + g'(t)j + h'(t)k

Tangent and Normal Vectors
Tangent Vector T(t)=r'(t)/abs(r'(t)), r'(t) != 0.

Normal Vector N(t)=T'(t)/abs(T'(t)).

Directional Derivative
If f is a differentiable function of x and y, then the directional derivatives of f in the direction of a unit vector u = cos θi + sin θj is

Duf(x,y) = fx(x,y)cosθ + fy(x,y)sinθ

For u = ai + bj + ck,

Duf(x,y,z) = afx(x,y,z) + bfy(x,y,z) + cfz(x,y,z);

Gradient
∇f(x,y,z) = fx(x,y,z)i + fy(x,y,z)j + fz(x,y,z)k

The direction of maximum increase is given by ∇f(x,y,z).

Related Rates

  1. Create a model (function) of the behavior
  2. Differentiate all variables with respect to rate (usually time)
  3. Plug in known values and rates

Extrema
To find the extrema of function f on closed interval [a,b]:

  1. Take the partial derivative of each variable (e.g. fx, fy fz) in f(x,y,z). Or find the gradient of the function f(x,y,z).
  2. Evaluate grad f=0. A critical number requires all partial derivatives to be zero at that coordinate.
  3. Evaluate f at each end point (or infinity).
  4. Evaluate f at each discontinuity.
  5. The minimum or maximum of the set will be the extrema. IOW extrema only occur at critical points.

Optimization: Standard

  1. Optimization Equation: the equation to be optimized (maximum or minimum).
  2. Boundary Equation(s): boundary condition equations.
  3. Make optimization equation a function of a single variable by substituting boundary equations into the optimization equation.
  4. Find extrema. Find derivative and solve for f(x)=0.
  5. Eliminate results that do not make sense. Check for minima and maxima.

Optimization: Lagrange

  1. Optimization Equation: the equation to be optimized (maximum or minimum).
  2. Boundary Equation(s): boundary condition equations.
  3. Take the partial derivative of the Optimization Equation for every variable in the equation (fx, fy, fz, etc).
  4. Take the partial derivative of the Boundary Equation for every variable in the equation (gx, gy, gz, etc).
  5. Create the Lagrange multipliers by the following equations: fi=λgi. These are the Lagrange multipliers.
  6. Solve the system of the Lagrange multipliers PLUS the Boundary Equation. This is a system of n+1 equations and n+1 variables, where n is the number of variables in the Optimization Equation.
  7. Plug the coordinate from the solution above into the Optimization Equation to find the optimum value.

Next Up
Integration

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