Differentiation is a mathematical process used to find the rate of change of a function with respect to its independent variable. It is often used to analyze the behavior of functions and can also be used to find maximum and minimum values of a function.
Differentiation and integration are inverse operations in calculus. Differentiation finds the rate of change of a function, while integration finds the accumulation of a function. In other words, differentiation is used to find the slope of a function, while integration is used to find the area under a function.
The basic rules for differentiating a function include the power rule, product rule, quotient rule, and chain rule. The power rule states that the derivative of x^n is n*x^(n-1), while the product rule states that the derivative of f(x)*g(x) is f'(x)*g(x) + f(x)*g'(x). The quotient rule states that the derivative of f(x)/g(x) is (f'(x)*g(x) - f(x)*g'(x))/g(x)^2. The chain rule is used to differentiate composite functions and states that the derivative of f(g(x)) is f'(g(x))*g'(x).
Differentiation is used in many areas of science and engineering, such as physics, chemistry, economics, and biology. It is used to analyze the behavior of systems and predict their future behavior. For example, in physics, differentiation is used to calculate the velocity and acceleration of objects, while in economics, it is used to analyze supply and demand curves.
Some common applications of differentiation include optimization problems, curve sketching, and related rates problems. Optimization problems involve finding the maximum or minimum value of a function, while curve sketching involves analyzing the behavior of a function by finding its critical points, asymptotes, and concavity. Related rates problems involve finding the rate of change of a quantity that is dependent on another changing quantity.