Calculus: Introduction to Derivatives

Introduction to Derivatives

• A derivative is a measure of how a function changes as its input changes.
• It is a fundamental concept in calculus, used to study the behavior of functions.

Notation and Terminology

• The derivative of a function f(x) is denoted as f'(x) or (d/dx)f(x).
• The process of finding a derivative is called differentiation.
• The derivative is a measure of the rate of change of the function with respect to its input.

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 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) and represents the rate of change of the first derivative.
• Higher-order derivatives can be used to study the concavity and inflection points of a function.

Applications of Derivatives

• Optimization: Derivatives are used to find the maximum and minimum values of a function.
• Physics: Derivatives are used to model the motion of objects, including the acceleration and velocity of an object.
• Economics: Derivatives are used to model the behavior of economic systems, including the rate of change of economic indicators.

Introduction to Derivatives

• A derivative measures how a function changes as its input changes and is a fundamental concept in calculus.
• It is used to study the behavior of functions.

Notation and Terminology

• The derivative of a function f(x) is denoted as f'(x) or (d/dx)f(x).
• The process of finding a derivative is called differentiation.
• The derivative measures the rate of change of the function with respect to its input.

Rules of Differentiation

• The Power Rule states that if f(x) = x^n, then f'(x) = nx^(n-1).
• The Product Rule states that if f(x) = u(x)v(x), then f'(x) = u'(x)v(x) + u(x)v'(x).
• The Quotient Rule states that if f(x) = u(x)/v(x), then f'(x) = (u'(x)v(x) - u(x)v'(x)) / v(x)^2.
• The Chain Rule states that 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 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) and represents the rate of change of the first derivative.
• Higher-order derivatives can be used to study the concavity and inflection points of a function.

Applications of Derivatives

• Derivatives are used in optimization to find the maximum and minimum values of a function.
• Derivatives are used in physics to model the motion of objects, including the acceleration and velocity of an object.
• Derivatives are used in economics to model the behavior of economic systems, including the rate of change of economic indicators.