**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 = f _{x}(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 | u^{n} dx = nu^{n-1}u' |

Sum/Difference | [u +/- v] dx = u' +/- v' |

Product | uv dx = uv' + vu' |

Quotient | (u/v) dx = (vu' - uv')/v^{2} |

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))**,

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

D_{u}f(x,y) = f_{x}(x,y)cosθ + f_{y}(x,y)sinθ

For *u = ai + bj + ck*,

D_{u}f(x,y,z) = af_{x}(x,y,z) + bf_{y}(x,y,z) + cf_{z}(x,y,z);

**Gradient***∇f(x,y,z) = f _{x}(x,y,z)i + f_{y}(x,y,z)j + f_{z}(x,y,z)k*

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

**Related Rates**

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

**Extrema**

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

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

**Optimization: Standard**

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

**Optimization: Lagrange**

- Optimization Equation: the equation to be optimized (maximum or minimum).
- Boundary Equation(s): boundary condition equations.
- Take the partial derivative of the Optimization Equation for every variable in the equation (f
_{x}, f_{y}, f_{z}, etc). - Take the partial derivative of the Boundary Equation for every variable in the equation (g
_{x}, g_{y}, g_{z}, etc). - Create the Lagrange multipliers by the following equations:
*f*_{i}=λg_{i}. These are the Lagrange multipliers. - 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.
- Plug the coordinate from the solution above into the Optimization Equation to find the optimum value.

**Next Up**

Integration

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