Calculus: Differentiation Rules

Differentiation

Definition

• Differentiation is a fundamental concept in calculus that deals with the rate of change of a function with respect to one of its variables.
• It is a measure of how a function changes as its input changes.

Notation

• The derivative of a function f(x) is denoted as f'(x) or (d/dx)f(x)
• The derivative of a function y = f(x) can also be denoted as dy/dx or y'

Rules of Differentiation

• Power Rule: If f(x) = x^n, then f'(x) = nx^(n-1)
• 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)

Geometric Interpretation

• The derivative of a function at a point represents the slope of the tangent line to the graph of the function at that point.
• The derivative can be used to find the maximum and minimum values of a function.

Higher-Order Derivatives

• The second derivative of a function f(x) is denoted as f''(x) or (d^2/dx^2)f(x)
• Higher-order derivatives can be used to find the concavity and inflection points of a function.

Applications of Differentiation

• Optimization: Differentiation is used to find the maximum and minimum values of a function, which is crucial in many fields such as economics, physics, and engineering.
• Physics: Differentiation is used to model the motion of objects, including the acceleration and velocity of particles and the curvature of space-time.
• Economics: Differentiation is used to model the behavior of economic systems, including the impact of changes in supply and demand on prices and quantities.

Differentiation

Definition

• Deals with the rate of change of a function with respect to one of its variables.
• Measures how a function changes as its input changes.

Notation

• Derivative of a function f(x) is denoted as f'(x) or (d/dx)f(x).
• Derivative of a function y = f(x) can also be denoted as dy/dx or y'.

Rules of Differentiation

• Power Rule: If f(x) = x^n, then f'(x) = nx^(n-1).
• 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).

Geometric Interpretation

• Derivative of a function at a point represents the slope of the tangent line to the graph of the function at that point.
• Derivative can be used to find the maximum and minimum values of a function.

Higher-Order Derivatives

• Second derivative of a function f(x) is denoted as f''(x) or (d^2/dx^2)f(x).
• Higher-order derivatives can be used to find the concavity and inflection points of a function.

Applications of Differentiation

• Optimization: Used to find the maximum and minimum values of a function, crucial in economics, physics, and engineering.
• Physics: Used to model the motion of objects, including acceleration and velocity of particles and curvature of space-time.
• Economics: Used to model the behavior of economic systems, including impact of changes in supply and demand on prices and quantities.