Limit Definition of a Derivative at a Point
The limit definition of a derivative at a point is the mathematical expression that describes how a function changes or behaves as the input value approaches a specific point
The limit definition of a derivative at a point is the mathematical expression that describes how a function changes or behaves as the input value approaches a specific point.
Let’s consider a function f(x) and a specific point ‘a’ at which we want to find the derivative. The derivative of f(x) at x=a is defined as:
f'(a) = lim(h→0) {f(a + h) – f(a)}/h
In this definition, h represents a small change in the x-coordinate (or input) around the point a. As h approaches zero, we take the limit to find the instantaneous rate of change, or the slope, of the function at point a.
To compute the derivative using this definition, follow these steps:
1. Identify the function f(x) for which you want to find the derivative.
2. Write out the limit expression: f'(a) = lim(h→0) {f(a + h) – f(a)}/h
3. Substitute the value of x=a into the expression: f'(a) = lim(h→0) {f(a + h) – f(a)}/h
4. Simplify the expression as much as possible.
5. Finally, take the limit as h approaches zero by evaluating the expression. This will provide the instantaneous rate of change, or the slope, of the function at point a.
It’s important to note that this limit definition provides a general approach to find the derivative at a specific value of x. However, in practice, there are other methods such as using differentiation rules or using derivatives of elementary functions to calculate derivatives more efficiently.
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