Introduction to Differentiation

Operation in calculus.

Differentiation is a fundamental concept in calculus that deals with rates of change. It is used to find the derivative of a function, which represents the rate at which the function is changing at any given point. This article will introduce the concept of differentiation, discuss the rules of differentiation, and explain how to differentiate various types of functions.

Definition of Derivative

The derivative of a function measures how the function changes as its input changes. In other words, it measures the rate of change of a function at a particular point. The derivative of a function f at a point x is denoted as f'(x) or df/dx.

Rules of Differentiation

There are several rules of differentiation that simplify the process of finding derivatives. These include:

1. Power Rule: The derivative of x^n, where n is any real number, is n*x^(n-1).
2. Product Rule: The derivative of a product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function.
3. Quotient Rule: The derivative of a quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.
4. Chain Rule: The derivative of a composition of functions is the derivative of the outer function times the derivative of the inner function.

Differentiating Various Types of Functions

Different types of functions require different techniques for differentiation:

• Polynomial Functions: The power rule is used to differentiate polynomial functions. Each term is differentiated separately, and the power rule is applied to each term.

• Rational Functions: The quotient rule is used to differentiate rational functions, which are functions that can be expressed as the quotient of two polynomials.

• Trigonometric Functions: Trigonometric functions have their own specific derivatives. For example, the derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x).

In conclusion, differentiation is a powerful tool in calculus that allows us to analyze the rate of change of functions. By understanding the rules of differentiation and how to apply them, we can find the derivatives of a wide range of functions.