Calculus: Differentiation

Study Notes

Differentiation is a process of finding the derivative of a function, which represents the rate of change of the function with respect to one of its variables.
It is a fundamental concept in calculus, used to study the behavior of functions, optimize functions, and model real-world phenomena.

Geometric Differentiation: finding the derivative of a function using geometric methods, such as tangents and slopes.
Limit-Based Differentiation: finding the derivative of a function using limits, which is the most common method.
Implicit Differentiation: finding the derivative of an implicitly defined function, where the function is defined implicitly using an equation.

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)

Finding the Maximum and Minimum Values: of a function, which is used in optimization problems.
Determining the Rate of Change: of a function, which is used to model real-world phenomena, such as population growth and motion.
Analyzing Functions: to determine the behavior of functions, such as identifying local maxima and minima, and inflection points.

Second Derivative: represents the rate of change of the first derivative, used to determine the concavity of a function.
Higher-Order Derivatives: represent the rate of change of lower-order derivatives, used to analyze functions and model complex phenomena.

Modeling Real-World Phenomena: differentiation is used to model real-world phenomena, such as population growth, motion, and optimization problems.
Optimization: differentiation is used to find the maximum and minimum values of a function, which is used in many fields, such as economics and engineering.
Analyzing Functions: differentiation is used to analyze functions and determine their behavior, which is used in many fields, such as physics and computer science.

Differentiation is a process of finding the derivative of a function, which represents the rate of change of the function with respect to one of its variables.
It is a fundamental concept in calculus, used to study the behavior of functions, optimize functions, and model real-world phenomena.

Geometric Differentiation: finding the derivative of a function using geometric methods, such as tangents and slopes.
Limit-Based Differentiation: finding the derivative of a function using limits, which is the most common method.
Implicit Differentiation: finding the derivative of an implicitly defined function, where the function is defined implicitly using an equation.

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)

Finding the Maximum and Minimum Values of a function, which is used in optimization problems.
Determining the Rate of Change of a function, which is used to model real-world phenomena, such as population growth and motion.
Analyzing Functions to determine the behavior of functions, such as identifying local maxima and minima, and inflection points.

Second Derivative: represents the rate of change of the first derivative, used to determine the concavity of a function.
Higher-Order Derivatives: represent the rate of change of lower-order derivatives, used to analyze functions and model complex phenomena.

Modeling Real-World Phenomena: differentiation is used to model real-world phenomena, such as population growth, motion, and optimization problems.
Optimization: differentiation is used to find the maximum and minimum values of a function, which is used in many fields, such as economics and engineering.
Analyzing Functions: differentiation is used to analyze functions and determine their behavior, which is used in many fields, such as physics and computer science.