Differentiation rules for functions from "summary" of Differential Calculus by S Balachandra Rao
Differentiation rules for functions are fundamental tools in calculus that allow us to find the derivative of a wide variety of functions. These rules provide a systematic way to differentiate functions without having to resort to first principles every time. By applying these rules, we can determine the rate of change of a function at any point, which is crucial in many real-world applications.
The power rule is one of the most basic differentiation rules and is used to find the derivative of a function raised to a constant power. For example, if we have a function f(x) = x^n, where n is a constant, then the derivative of f(x) with respect to x is n*x^(n-1). This rule simplifies the process of finding derivatives of polynomials and other functions with exponents.
Another important rule is the sum rule, which states that the derivative of a sum of functions is equal to the sum of the derivatives of the individual functions. This rule allows us to break down complex functions into simpler parts and differentiate them separately. Similarly, the product rule and quotient rule provide guidelines for finding the derivatives of products and quotients of functions, respectively.
The chain rule is another crucial differentiation rule that allows us to find the derivative of composite functions. If we have a function g(x) = f(h(x)), where h(x) is a function of x and f(x) is a function of h(x), then the chain rule states that the derivative of g(x) with respect to x is equal to the derivative of f with respect to h multiplied by the derivative of h with respect to x.
In addition to these basic rules, there are other differentiation rules for trigonometric, exponential, and logarithmic functions. These rules provide shortcuts for finding the derivatives of these common functions, saving time and effort in the differentiation process. By mastering these differentiation rules, we can efficiently compute derivatives and analyze the behavior of functions in various contexts.
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