Basic Derivative
f(x^n)= nX^(n-1)
The derivative of a function is a measure of how the function changes as its input (independent variable) changes. Specifically, the derivative of a function f(x) with respect to its input x, denoted as f'(x), is defined as the limit of the rate of change of the function as the change in input (denoted as h) approaches zero. Mathematically, this can be expressed as:
f'(x) = lim(h->0) [f(x+h) – f(x)] / h
Intuitively, the derivative of a function represents the slope of a tangent line to the function at a given point, which describes how quickly the function is changing at that point. To find the derivative of a function, we can use several methods, including the power rule, product rule, quotient rule, and chain rule, among others. It’s important to note that not all functions have derivatives, and some functions may have derivatives only in certain intervals or at discontinuities. Also, derivatives can have many applications in mathematics and science, including optimization, modeling, and physics.
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