In mathematics, differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to its input variable. The derivative of a function f(x) with respect to x is denoted as f'(x) or df/dx.
The Class 12 Mathematics syllabus covers the following topics related to differentiation:
Definition and Meaning of Derivative: This topic introduces the concept of derivative, its definition, and interpretation as the rate of change of a function.
Derivatives of Standard Functions: This topic covers the derivatives of standard functions such as polynomial, trigonometric, exponential, and logarithmic functions.
Techniques of Differentiation: This topic covers the various techniques of differentiation, such as the product rule, quotient rule, chain rule, and implicit differentiation.
Derivatives of Functions Expressed in Parametric Forms: This topic covers the differentiation of functions expressed in parametric forms.
Second Order Derivatives: This topic covers the concept of second-order derivatives, their interpretation as the rate of change of the rate of change of a function, and their applications in curve sketching.
The study of differentiation is important because it is a fundamental concept in calculus and is used in various fields such as physics, engineering, economics, and statistics. It provides a powerful tool for calculating rates of change, slopes of curves, and optimization of functions. The techniques of differentiation are used in solving problems related to optimization, related rates, curve sketching, and integration.