Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Derivatives and Continuity

Description

This lecture covers the concept of continuous differentiability, the Bernoulli-l'Hôpital rule, and the application of derivatives to find extrema. It also explores the proof of the rule of Bernoulli-l'Hôpital and the importance of continuous differentiability in finding extrema.

Official source

Ontological neighbourhood

Mathematics

Analysis: Calculus

Related concepts (32)

Differentiable function

In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp. If x0 is an interior point in the domain of a function f, then f is said to be differentiable at x0 if the derivative exists.

Semi-differentiability

In calculus, a branch of mathematics, the notions of one-sided differentiability and semi-differentiability of a real-valued function f of a real variable are weaker than differentiability. Specifically, the function f is said to be right differentiable at a point a if, roughly speaking, a derivative can be defined as the function's argument x moves to a from the right, and left differentiable at a if the derivative can be defined as x moves to a from the left.

Derivative

In mathematics, the derivative shows the sensitivity of change of a function's output with respect to the input. Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.

Differentiation rules

This is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus. Unless otherwise stated, all functions are functions of real numbers (R) that return real values; although more generally, the formulae below apply wherever they are well defined — including the case of complex numbers (C). For any value of , where , if is the constant function given by , then . Let and . By the definition of the derivative, This shows that the derivative of any constant function is 0.

Mathematical proof

A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation".

Related lectures (39)

Chain Rules

Explores the extension of the chain rule to higher dimensions and compositions of functions.

Derivatives and Reciprocal Functions

Covers derivatives, reciprocal functions, Rolle's theorem, and extremum local concepts.

Demonstration of Rolle's Theorem MOOC: Analysis I MOOC: Analysis I (part 6) : Study of functions, limited developments

Demonstrates how to use a specific function to prove Rolle's Theorem.

Derivatives and Local Extrema

Explores derivatives, local extrema, and function variation in mathematical analysis.

Taylor Series: Expansion and Properties