Alternate Definition of Derivative
limit (as x approaches a number c)=f(x)-f(c)/x-c x≠c
The derivative of a function is an alternate definition that describes the rate at which the function changes with respect to its input variable. It can also be described as the slope of the tangent line to the graph of the function at a given point.
To calculate the derivative of a function f(x) at a specific point x=a, we can use the following formula:
f'(a) = lim (h -> 0) [f(a + h) – f(a)]/h
where h represents a very small change in the input variable around the point a. Essentially, this formula calculates the limit of the difference quotient as h approaches 0, giving us the instantaneous rate of change of the function at the point a.
The derivative is used extensively in calculus to find critical points, determine the concavity of curves, and solve optimization problems. It is also a fundamental concept in physics, where it is used to describe the rate of change of quantities such as velocity and acceleration.
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