Limit Definition of Derivative
f'(x) = lim h→0 ([f(x+h)-f(x)]/h)
The limit definition of derivative is a mathematical expression that defines the derivative of a function at a given point. It is the formal definition of the derivative and is one of the most important concepts in calculus.
The limit definition of derivative is given by:
f'(a) = lim[h→0] (f(a + h) – f(a)) / h
where a is the point at which the derivative is to be calculated, h is a small change in the x-value around the point a, f(a) is the value of the function at the point a, and f(a+h) is the value of the function after a small change h in the x-value.
This definition represents the slope of the tangent line to the function at the point a on the graph. By taking the limit as h approaches zero, we can obtain an exact value for the slope of the tangent line and thus the instantaneous rate of change of the function at the point a.
In simpler terms, the limit definition of derivative is used to calculate the slope of a curve at a specific point. It is a powerful concept that forms the basis of many of the applications of calculus, including optimization, optimization problems, and curve sketching.
More Answers:
Master the Product Rule for Differentiation: A Fundamental Tool for Optimization, Physics, and EngineeringThe Power Rule in Calculus: Deriving Functions Raised to Any Power
Understanding the Limit Definition of Derivatives: A Fundamental Concept in Calculus