derivatives

• What Is the Derivative of an Indicator Function?
  An indicator function has classical derivative zero where it is locally constant and no derivative at jump points; distributionally, jumps become Dirac deltas.
• How to Differentiate a Double Summation Using Dot Notation
  For a finite double sum with fixed limits, differentiate each term; dot notation means a derivative with respect to time, not a special summation rule.
• What Is the Derivative of an Indicator Function?
  An indicator function has ordinary derivative zero wherever it is locally constant; at jumps the derivative fails to exist, or is a delta distribution.
• What Is the Derivative of the Sigmoid Function?
  For the logistic sigmoid σ(x)=1/(1+e^−x), the derivative is σ(x)(1−σ(x)); see the derivation, chain rule, and worked examples.
• How Do You Differentiate a Triangle Wave?
  Differentiate a triangle wave by finding the slope of each linear segment; learn why its derivative is a square wave and is undefined at corners.
• How Do You Differentiate a Double Summation?
  Differentiate finite double sums term by term; for infinite sums, check uniform convergence of the derivative series before interchanging differentiation and summation.
• What Is the Derivative of an Indicator Function, and Its Integral?
  An interval's indicator has ordinary derivative zero except at its jumps; its integral is overlap length, and its distributional derivative records those jumps.
• How to Find the Derivative of the Logistic Function
  Differentiate the logistic sigmoid using the chain rule, obtain σ′(x) = σ(x)(1 − σ(x)), and adapt the result to scaled logistic functions.