Understanding the Formal Definition of Derivative in Mathematics

Formal definition of derivative

The formal definition of the derivative of a function can be expressed as follows:

Let f(x) be a function defined on an interval containing a point a

The formal definition of the derivative of a function can be expressed as follows:

Let f(x) be a function defined on an interval containing a point a. The derivative of f at the point a, denoted as f'(a) or dy/dx evaluated at a, is defined as the limit:

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

This definition represents the rate of change of the function f(x) at a specific point a. It calculates the slope of a tangent line to the function’s graph at that point.

The limit as h approaches 0 signifies that we are considering the change in the function as h becomes increasingly small. By subtracting the value of the function at a from the value of the function at a+h, and dividing this difference by h, we obtain an approximation of the slope of the secant line passing through the points (a, f(a)) and (a+h, f(a+h)). Taking the limit gives us the exact slope of the tangent line at the point a.

This formal definition applies to all differentiable functions, meaning those for which the derivative exists at every point in their domain.

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