Calculus: Differentiation Rules and Formulas

• Differentiation determines the rate of change of a function.
• Differentiation geometrically represents the slope of the tangent line to a function's graph at a point.

Rules of Differentiation

• Power rule: If f(x) = x^n, then f'(x) = nx^(n-1).
• Constant multiple rule: If f(x) = cg(x), then f'(x) = cg'(x), where c is a constant.
• Sum/difference rule: If f(x) = u(x) ± v(x), then f'(x) = u'(x) ± v'(x).
• Product rule: If f(x) = u(x)v(x), then f'(x) = u'(x)v(x) + u(x)v'(x).
• Quotient rule: If f(x) = u(x)/v(x), then f'(x) = [u'(x)v(x) - u(x)v'(x)] / [v(x)]^2.
• Chain rule: If f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).

Derivatives of Trigonometric Functions

• The derivative of sin(x) is cos(x).
• The derivative of cos(x) is -sin(x).
• The derivative of tan(x) is sec^2(x).
• The derivative of cot(x) is -csc^2(x).
• The derivative of sec(x) is sec(x)tan(x).
• The derivative of csc(x) is -csc(x)cot(x).

Derivatives of Exponential and Logarithmic Functions

• The derivative of e^x is e^x.
• The derivative of a^x is a^x * ln(a).
• The derivative of ln(x) is 1/x.
• The derivative of log_a(x) is 1 / (x * ln(a)).

Higher-Order Derivatives

• Higher-order derivatives are derivatives of derivatives.
• The second derivative, denoted as f''(x) or d^2y/dx^2, is the derivative of the first derivative f'(x).
• The nth derivative, denoted as f^(n)(x) or d^ny/dx^n, is the derivative of the (n-1)th derivative.
• Higher-order derivatives provide information about the concavity and rate of change of the rate of change of a function.

Differential Equations

• A differential equation relates a function with its derivatives.
• Ordinary differential equations (ODEs) involve functions of one independent variable.
• Partial differential equations (PDEs) involve functions of several independent variables and their partial derivatives.
• The order of a differential equation is the highest order derivative in the equation.
• The degree of a differential equation is the power of the highest order derivative, after the equation has been rationalized.

Types of Differential Equations

• First-order differential equations involve only the first derivative.
• Second-order differential equations involve the second derivative.
• Linear differential equations are those in which the dependent variable and its derivatives appear linearly.
• Homogeneous differential equations are those in which all terms have the same degree.
• Separable differential equations can be written in the form dy/dx = f(x)g(y).

Applications in Physics

• Kinematics: Differentiation finds velocity and acceleration from displacement.
  • Velocity is the derivative of displacement with respect to time: v = ds/dt.
  • Acceleration is the derivative of velocity with respect to time: a = dv/dt = d^2s/dt^2.
• Newton's Second Law: F = ma, where acceleration is the second derivative of position with respect to time.
• Simple Harmonic Motion (SHM): The equation of motion involves second derivatives, representing oscillatory behavior.
• Wave equations describe the propagation of waves and involve partial derivatives with respect to time and space.
• Heat equation describes how temperature changes over time in a given region.
• Schrödinger equation is a fundamental equation in quantum mechanics that describes the time evolution of a quantum system.

Partial Differentiation

• Partial differentiation differentiates a function of several variables with respect to one variable, holding the other variables constant.
• Notation: If f(x, y) is a function of x and y, then ∂f/∂x represents the partial derivative of f with respect to x, and ∂f/∂y represents the partial derivative of f with respect to y.
• To find ∂f/∂x, treat y as a constant and differentiate f(x, y) with respect to x.
• To find ∂f/∂y, treat x as a constant and differentiate f(x, y) with respect to y.

Higher-Order Partial Derivatives

• Second-order partial derivatives include ∂^2f/∂x^2, ∂^2f/∂y^2, ∂^2f/∂x∂y, and ∂^2f/∂y∂x.
• ∂^2f/∂x^2 is found by differentiating ∂f/∂x with respect to x.
• ∂^2f/∂y^2 is found by differentiating ∂f/∂y with respect to y.
• ∂^2f/∂x∂y is found by differentiating ∂f/∂y with respect to x.
• ∂^2f/∂y∂x is found by differentiating ∂f/∂x with respect to y.
• Clairaut's Theorem: If the second partial derivatives are continuous, then ∂^2f/∂x∂y = ∂^2f/∂y∂x.

Applications of Partial Differentiation

• Optimization: Finding maximum and minimum values of functions of several variables.
• Physics: Thermodynamics, fluid dynamics, and electromagnetism.
• Engineering: Analyzing stress and strain in materials, heat transfer, and fluid flow.
• Economics: Modeling economic behavior and optimizing resource allocation.