Limit Definition of Derivative
The limit definition of derivative is a mathematical expression that describes how the derivative of a function at a specific point can be determined using limits
The limit definition of derivative is a mathematical expression that describes how the derivative of a function at a specific point can be determined using limits.
Let’s say we have a function f(x) and we want to find its derivative at a point x = a. The limit definition of the derivative is given by:
f'(a) = lim (h -> 0) [f(a + h) – f(a)] / h
In this expression, h represents a small change in x. We take the limit as h approaches 0 because we want to evaluate the instantaneous rate of change of the function at x = a.
To compute the derivative using the limit definition, we follow these steps:
1. Start with the given equation: f'(a) = lim (h -> 0) [f(a + h) – f(a)] / h.
2. Substitute the function f(x) with its actual expression.
3. Replace x with (a + h) in the numerator.
4. Expand and simplify the expression in the numerator, which will give us f(a + h) – f(a).
5. Divide the resulting expression by h.
6. Take the limit as h approaches 0.
Evaluating this limit will give us the value of the derivative at the point x = a.
It’s important to note that the limit definition of derivative provides the foundation for many fundamental concepts in calculus, such as the power rule, product rule, chain rule, and more. Using this definition, we can calculate the derivative of any function at any point.
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