Understanding the Derivative in Calculus | An Alternate Definition Using Limits

Alternate definition of derivative

In calculus, the derivative of a function at a certain point is a measure of how the function changes at that point

In calculus, the derivative of a function at a certain point is a measure of how the function changes at that point. An alternate definition of the derivative can be given using the concept of limits.

Let’s consider a function f(x) and a particular point ‘a’. The derivative of f(x) at ‘a’ can be defined using the limit:

f'(a) = lim (h –> 0) [(f(a + h) – f(a)) / h]

Here, ‘h’ represents a small change in the x-coordinate from ‘a’. The derivative f'(a) represents the rate of change of f(x) at the specific point ‘a’ and can be interpreted as the slope of the tangent line to the graph of f(x) at that point.

This alternate definition essentially calculates the average rate of change of f(x) over an interval, as ‘h’ approaches zero. By taking the limit, we essentially shrink the interval to an infinitesimally small size, allowing us to determine the instantaneous rate of change at the specific point ‘a’.

It is important to note that this alternate definition applies to functions that are differentiable at the point ‘a’, which means that the function has a defined slope at that point and a tangent line can be drawn without any discontinuities or sharp turns.

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