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Calculus: Derivatives (Part 3 of x)

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.

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.


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 Limits

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

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)).

Related Rates

  1. Sketch function
  2. Differentiate all variables with respect to rate (usually time)
  3. Plug in known values and rates

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).
  2. Find the critical numbers of each partial derivative in the previous step (fx=0).
  3. Evaluate f at each critical number.
  4. Evaluate f at each end point.
  5. The minimum or maximum of the set will be the extrema.


  1. Optimization Equation: the equation to be optimized (maximize or minimize).
  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.

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);

∇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).

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