Derivatives

• How to Differentiate f(x) = x√(2x − 3)
  Differentiate x√(2x − 3) using the product and chain rules, simplify the result, and see where the derivative is defined.
• How to Find the Gradient Using Differentiation
  Find the gradient of a curve by differentiating its function and evaluating the derivative at the required point; see a worked example.
• How to Find a Derivative Using the Difference Quotient
  Use the difference quotient [f(x+h) − f(x)]/h, simplify before taking its limit as h approaches zero, and interpret the result as the derivative.
• How to Find the Rates of Change of a Rectangle’s Area, Perimeter, and Diagonal
  Find the instantaneous rates for a rectangle with changing sides: area rate, perimeter rate, and the common rate of its two equal diagonals.
• How to Find the Area in a Ladder Related-Rates Problem
  For a sliding ladder, model the wall, floor, and ladder as a right triangle, then differentiate its area and the fixed-length constraint.
• Can You Find a Function from Its Tangent Line?
  A tangent line fixes a function's value and derivative at one point, but usually does not determine the function without extra conditions.
• How to Derive Demand from Utility and an Endowment
  Derive a consumer's demand by valuing the endowment, maximizing utility subject to the budget, and distinguishing chosen consumption from net trade.
• What Do You Get When You Differentiate a Function Twice?
  Differentiating twice gives the second derivative in one variable and the Hessian matrix for a scalar function of several variables.
• How to Differentiate a Piecewise Function
  Differentiate each formula on its interval, then test every breakpoint with continuity and matching one-sided derivatives to determine whether the derivative exists.
• How to Differentiate Bessel Functions
  Differentiate Bessel functions using recurrence identities, with formulas for J, Y, I, and K and a worked example involving a weighted function.
• Does $f(0)=0$ Imply That $f'(x)=0$?
  Knowing that f(0)=0 tells you only that the graph passes through the origin; it does not determine whether f' exists or what its value is.
• How to Prove a Graph Has No Horizontal Tangent Lines
  To prove a differentiable graph has no horizontal tangents, show its derivative is never zero on the relevant domain; the function is needed for a specific proof.
• How Do You Find the Slope of a Function?
  Find a function’s slope by differentiating it, then evaluate the derivative at the input you care about. The result is the tangent-line slope.
• Can a Function Have First and Second Derivatives Equal to Zero?
  Yes. For example, f(x) = x^3 has f'(0) = 0 and f''(0) = 0; the same conditions can also occur at a local minimum, as with x^4.
• What Is the Approximation Formula for a Higher-Order Derivative?
  Approximate an nth derivative with finite differences, understand the error, and see a worked second-derivative example using a central stencil.
• Are Directional Derivatives Scalars or Vectors?
  A directional derivative is scalar for a scalar-valued function and vector-valued for a vector-valued function; the direction itself is an input vector.
• 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.
• 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.