The derivative of a function at a point measures the rate at which the function's value changes as the function's argument changes. That is, a derivative provides a mathematical formulation of the notion of rate of change. As it turns out, the derivative is an extremely versatile concept which can be viewed in many different ways. For example, referring to the two-dimensional graph of f, the derivative can also be regarded as the slope of the tangent to the graph at the point x. The slope of this tangent can be approximated by a secant. Given this geometrical interpretation, it is not surprising that derivatives can be used to determine many geometrical properties of graphs of functions, such as concavity or convexity.

It should be noted that not all functions have derivatives. For example, functions do not have derivatives at points where they have either a vertical tangent or a discontinuity. However, functions may fail to have derivatives even if they are continuous and have no vertical tangents.




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