Understanding the Limit Definition of the Derivative | Calculating Instantaneous Rate of Change in Mathematics

Limit Definition of Derivative

The limit definition of the derivative is a mathematical formula that allows us to calculate the rate at which a function is changing at any given point

The limit definition of the derivative is a mathematical formula that allows us to calculate the rate at which a function is changing at any given point. It describes the instantaneous rate of change of a function at a specific point.

Let’s start with a function, f(x), and a specific point on the function, let’s call it ‘a’. The derivative of f(x) at this point, denoted as f'(a) or dy/dx (read as “dy by dx”), can be defined using the limit as follows:

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

Here’s how it works:
1. We take a small interval around ‘a’ by adding a small value ‘h’ to ‘a’. This creates a new point (a + h) on the function.
2. We calculate the difference between the function values at these two points, f(a + h) – f(a). This represents the change in the function over the interval from a to (a + h).
3. We divide this difference by ‘h’, the size of the interval. This gives us the average rate of change over the interval.
4. Finally, we take the limit as ‘h’ approaches zero, which allows the interval to shrink to an infinitesimally small size. This gives us the instantaneous rate of change, or the derivative, at point ‘a’.

In summary, the limit definition of the derivative evaluates the change in a function over a small interval as the interval size approaches zero. This concept is used to calculate derivatives of functions and plays a crucial role in the study of calculus.

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