First Principles of Differentiation Differentiation is the process of finding the derivative of a function, which is a measure of how quickly the function c...
First Principles of Differentiation
Differentiation is the process of finding the derivative of a function, which is a measure of how quickly the function changes with respect to its input.
Fundamental Theorem of Calculus
The fundamental theorem of calculus states that the derivative of a function f(x) at any point x = a is equal to the limit of the difference quotient as h approaches 0:
Definition of the Derivative
The derivative of a function f(x) at a point x = a is defined as the instantaneous rate of change of f(x) with respect to x at that point.
Derivation
The derivative of a function f(x) is found by taking the limit of the difference quotient as h approaches 0. This limit represents the instantaneous rate of change of f(x) at x = a.
Key Concepts
The derivative of a function at a given point is a single numerical value.
The derivative of a function can be found by applying the fundamental theorem of calculus.
The derivative of a function is a function itself.
The derivative of a composite function is equal to the derivative of the outer function evaluated at the inner function.
Examples
Consider the function f(x) = x^2. The derivative of this function is 2x.
Consider the function f(x) = sin(x). The derivative of this function is cos(x).
Applications of Differentiation
The first principles of differentiation have numerous applications in various fields, including mathematics, physics, economics, and engineering. They are used to solve problems involving rates of change, optimization, and related rates