Alternate definition of derivative
limit as x approaches a of [f(x)-f(a)]/(x-a)
One alternate definition of derivative is that it represents the instantaneous rate of change of a function at a specific point. This means that as the input (independent variable) approaches the point in question, the rate at which the output (dependent variable) is changing is captured by the derivative of the function at that point.
For example, if we have a function f(x) = x^2, the derivative of f(x) at x = 3 represents the rate at which the value of f(x) is changing at x = 3. Graphically, this is represented by the slope of the tangent line to the graph of f(x) at x = 3.
This alternate definition of derivative is closely related to the concept of limits, as the derivative is defined as the limit of the difference quotient as the change in input approaches zero. The derivative can also be thought of as the limit of the average rate of change of the function on an interval around the point in question, as the interval becomes smaller and smaller.
Overall, the alternate definition of derivative as the instantaneous rate of change provides an intuitive understanding of the concept for both practical and theoretical applications in calculus and beyond.
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