Alternate definition of derivative at x=a
In Calculus, the derivative of a function at a specific point is typically defined as the slope of the line tangent to the graph of the function at that point
In Calculus, the derivative of a function at a specific point is typically defined as the slope of the line tangent to the graph of the function at that point. However, there is another alternate definition of the derivative at a specific point, known as the limit definition of the derivative.
The alternate definition of the derivative at x=a is given by the following limit:
f'(a) = lim┬(h→0)((f(a + h) – f(a))/h)
Let’s break down this definition and understand its meaning:
1. f(a + h) represents the value of the function f at the point (a + h). This means we are considering a small interval around the point a, where h represents the distance from a.
2. f(a) represents the value of the function f at the point a. This is the y-value of the point on the graph where we want to find the derivative.
3. (f(a + h) – f(a)) represents the change in y-values between the points (a + h) and a. It measures the rise of the graph.
4. h represents the change in x-values between the points (a + h) and a. It measures the run of the graph.
5. The division (f(a + h) – f(a))/ h calculates the slope of the secant line passing through the points (a + h, f(a + h)) and (a, f(a)).
6. Finally, taking the limit as h approaches 0 ensures that the secant line coincides with the tangent line at the point a, giving us the slope of the tangent line.
By using this limit definition of the derivative, we can find the derivative of a function at any given point. It is especially useful when the function is not differentiable or when the usual derivative rules are not applicable.
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