Definition of the Derivativef'(x) = ___________________
The derivative of a function, denoted as f'(x), represents the rate at which the function is changing at a particular point
The derivative of a function, denoted as f'(x), represents the rate at which the function is changing at a particular point. It measures the slope or steepness of the function’s graph at that specific point.
Mathematically, the derivative of a function f(x) can be defined as the limit of the difference quotient as the interval around a point x approaches zero. The difference quotient is the ratio of the change in the y-values of the function divided by the change in the x-values.
So, the general formula for the derivative of a function f(x) is:
f'(x) = lim(h -> 0) [(f(x + h) – f(x)) / h]
This equation essentially calculates the slope of the tangent line to the graph of the function f(x) at the point (x, f(x)). The limit as h approaches zero ensures that the derivative provides the instantaneous rate of change at that specific point.
The derivative can be interpreted as the rate of change of the function with respect to x, the slope of the tangent line to the graph of the function, or the velocity of an object moving along a curve represented by the function.
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