Basic Derivative
f(x^n)= nX^(n-1)
A derivative is a mathematical concept that describes the rate at which a function is changing at a given point on its graph. In calculus, it is defined as the limit of the ratio of the change in output (y-value) of a function to the change in input (x-value) as the change in input approaches zero. The result of this calculation is the slope or gradient of the function at that point.
To find the derivative of a function, we use differentiation rules, also known as derivative rules or formulae. These rules provide us with a useful framework for calculating the derivatives of various functions.
The basic derivative rules include:
1. Power Rule: the derivative of a function xn is nxn-1.
2. Product Rule: the derivative of the product of two functions, f(x) and g(x), is f'(x)g(x)+f(x)g'(x).
3. Quotient Rule: the derivative of the quotient of two functions, f(x) and g(x), is (f'(x)g(x)-f(x)g'(x))/g(x)².
4. Chain Rule: the derivative of a composite function f(g(x)) is f'(g(x))g'(x).
5. Sum or Difference Rule: the derivative of the sum or difference of two functions, f(x) and g(x), is f'(x)+g'(x) and f'(x)-g'(x), respectively.
These derivative rules, when applied systematically and correctly, can be used to determine the derivative of any function. Derivatives have many applications in mathematics, physics, and engineering, including optimization, curve sketching, and solving differential equations.
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