"The derivative shows the sensitivity of change of a function's output with respect to the input."
The derivative of a function is the rate at which the function is changing at any given point.
Limits: The concept of a limit is used to define derivatives in calculus.
Continuity: Continuity is an important concept to understand when learning about derivatives as they represent how functions change over time.
Differentiation: The process of finding derivatives of functions and understanding their properties.
Rules of Differentiation: There are several rules of differentiation that make the process of finding derivatives easier.
Chain Rule: The chain rule is a calculus technique used to differentiate composite functions.
Implicit Differentiation: This is used when a function is not given in explicit form but instead is represented implicitly through an equation.
Related Rates: Related rates is a method to find the rate of change of one quantity in terms of the rate of change of another.
Applications of Differentiation: The applications of derivatives in real-life situations, including optimization and related rates problems.
Higher Derivatives: What happens when we differentiate a function more than once?.
Curve Sketching: Understanding the shape of curves helps in understanding their derivatives and vice versa.
Differentiation of Inverses: A technique used to find the derivative of a function that is the inverse of another function.
Mean Value Theorem: A theorem that relates the average rate of change of a function to its derivative at a specific point.
Optimization: Determining the maximum or minimum value of a function.
Integration: Integration is the inverse process of differentiation and is also important in the study of derivatives.
Differential Equations: Differential equations are equations that involve derivatives and are used in various fields like physics, engineering, economics, etc.
Power Rule: Differentiation of a variable raised to a power.
Product Rule: The differentiation of the product of two functions.
Quotient Rule: The differentiation of the quotient of two functions.
Chain Rule: The differentiation of a function within a function.
Implicit Differentiation: Differentiation of the dependent variable with respect to the independent variable in an implicitly defined function.
Logarithmic Differentiation: A way to take the derivative of functions that contain products, quotients, and powers using logarithms.
Trigonometric Functions: Differentiation of trigonometric functions such as sin, cos, tan, etc.
Exponential Functions: Differentiation of exponential functions such as e^x.
Inverse Functions: Differentiation of inverse functions such as arc tan, arc sin, etc.
Parametric Functions: Differentiation of functions that are defined parametrically.
Vector Calculus: Differentiation of vector functions.
Partial Derivatives: The differentiation of a function with respect to one of its variables while keeping the other variable(s) constant.
"The derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances."
"The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point."
"For this reason, the derivative is often described as the 'instantaneous rate of change', the ratio of the instantaneous change in the dependent variable to that of the independent variable."
"Derivatives can be generalized to functions of several real variables."
"The derivative is reinterpreted as a linear transformation whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function."
"The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables."
"It can be calculated in terms of the partial derivatives with respect to the independent variables."
"For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector."
"The process of finding a derivative is called differentiation."
"The reverse process is called antidifferentiation."
"The fundamental theorem of calculus relates antidifferentiation with integration."
"Differentiation and integration constitute the two fundamental operations in single-variable calculus."
"The derivative shows the sensitivity of change of a function's output with respect to the input."
"The derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances."
"The tangent line is the best linear approximation of the function near that input value."
"The derivative is often described as the 'instantaneous rate of change'."
"Derivatives can be generalized to functions of several real variables."
"The Jacobian matrix is the matrix that represents this linear transformation."
"Differentiation and integration constitute the two fundamental operations in single-variable calculus."