Mastering Calculus: Understanding the Limit Definition of Derivative

Limit Definition of Derivative

limit (as h approaches 0)= F(x+h)-F(x)/h

The limit definition of derivative is a mathematical equation that describes the slope of a curve at a specific point. It is the fundamental formula used in calculus to determine derivatives. Let’s consider a function f(x) and a point ‘a’ on its graph. The slope of the curve at that point ‘a’ is defined as the derivative of f(x) at that point. The limit definition of derivative is given by,

f'(a) = lim h→0 (f(a+h) – f(a))/h

Here, h is a very small value that approaches zero but is not exactly zero. It represents the distance between the x-value ‘a’ and the x-value ‘a + h’ on the curve. When h is close to zero, it signifies that we are considering the point ‘a’ and the point ‘a+h’ to be very close to each other, almost like two points merged into one. This means that we are looking at the slope of the curve at the immediate vicinity of ‘a’.

The above equation tells us that to find the slope of a curve at a point ‘a’, we should take the limit of the difference quotient as h approaches zero. We can find the derivative of a function f(x) at any point ‘a’ on its graph using this equation. This concept is fundamental to calculus, and it is used in various fields such as physics, engineering, economics, and others.

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