Definition: In mathematics, a derivative represents the rate at which a function is changing at any given point. It measures how a function's output value changes as the input changes.

• Notation:
  • $f'(x)$ or $\frac{dy}{dx}$ for the derivative of $y$ with respect to $x$.

  • Basic Concepts

    • Function: A relationship between a set of inputs and a set of permissible outputs.
    • Rate of Change: The speed at which a variable changes over a specific period of time.

    Calculating Derivatives

    • First Principle (Definition of Derivative):
      • The derivative of $f(x)$ at $x$ is defined as: $$f'(x) = \lim_{{h \to 0}} \frac{f(x + h) - f(x)}{h}$$

  • Rules of Differentiation

    • Power Rule: If $f(x) = x^n$, then $f'(x) = nx^{n-1}$
      
    • Sum Rule: If $f(x) = g(x) + h(x)$, then $f^{\mathrm{\prime }}(x)=g^{\mathrm{\prime }}(x)+h^{\mathrm{\prime }}(x)$
    • Difference Rule: If $f(x) = g(x) - h(x)$, then $f'(x) = g'(x) - h'(x)$
      
    • Product Rule: If $f(x) = g(x) \cdot h(x)$, then $f^{\mathrm{\prime }}(x)=g^{\mathrm{\prime }}(x)h(x)+g(x)h^{\mathrm{\prime }}(x)$
    • Quotient Rule: If $f(x) = \frac{g(x)}{h(x)}$, then $f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}$
      
    • Chain Rule: If $f(x) = g(h(x))$, then $f'(x) = g'(h(x)) \cdot h'(x)$
      

    Higher-Order Derivatives

    • Second Derivative: The derivative of the derivative of a function, denoted as $f''(x)$ or $\frac{d^{2}y}{d{x}^2}$
    • Applications: Used to determine the concavity of a function and find points of inflection.

    Applications in Business and Economics

    • Marginal Analysis:
      • Marginal Cost: The derivative of the total cost function with respect to the quantity of output.
      • Marginal Revenue: The derivative of the total revenue function with respect to the quantity of output.
    • Optimization:
      • Profit Maximization: Finding the quantity of output where marginal cost equals marginal revenue.
      • Cost Minimization: Determining the point at which the cost is minimized.

    Implicit Differentiation

    • Definition: A method to find the derivative when the function is given in an implicit form (not solved for one variable in terms of another).
    • Process:
      1. Differentiate both sides of the equation with respect to $x$.
      2. Solve for $\frac{dy}{dx}$.

    Related Rates

    • Definition: Techniques for finding the rates at which two or more related variables change with respect to time.
    • Procedure:
      1. Identify the variables that change with time.
      2. Write an equation relating these variables.
      3. Differentiate both sides of the equation with respect to time.
      4. Solve for the desired rate.

    Derivatives of Exponential and Logarithmic Functions

    • Exponential Functions: If $f(x) = e^x$, then $f'(x) = e^x$.
    • Logarithmic Functions: If $f(x)=\ln (x)<span\; style="font-family:\; sans-serif;\; text-align:\; var(--bs-body-text-align);">,\; then </span>$$f'(x) = \frac{1}{x}$
      

    Practical Examples

    • Demand Function: Analyzing how the demand for a product changes with respect to price.
    • Supply Function: Understanding the relationship between the quantity supplied and the price.
    • Elasticity of Demand: Using derivatives to measure the responsiveness of the quantity demanded to changes in price.

    Techniques for Finding Derivatives

    • Substitution: Simplifying functions by substituting variables.
    • Differentiating Inverse Functions: Finding derivatives of inverse functions by using the formula $(f^{-1})'(y) = \frac{1}{f'(x)}$ where $y = f(x)$.