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Lecture# Derivability and Chain Rule

Description

This lecture covers the demonstration of the chain rule, including the derivative of composite functions and the proof of the theorem. It also explores the concept of derivability and the demonstration of the theorem of Rolle.

Official source

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In course

MATH-101(d): Analysis I

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles d'une variable.

Related concepts (217)

Gamma function

In mathematics, the gamma function (represented by Γ, the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer n, Derived by Daniel Bernoulli, for complex numbers with a positive real part, the gamma function is defined via a convergent improper integral: The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple poles.

Transcendental function

In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function. In other words, a transcendental function "transcends" algebra in that it cannot be expressed algebraically. Examples of transcendental functions include the exponential function, the logarithm, and the trigonometric functions. Formally, an analytic function f (z) of one real or complex variable z is transcendental if it is algebraically independent of that variable.

Trigonometric functions

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.

Continuous function

In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is .

Uniform continuity

In mathematics, a real function of real numbers is said to be uniformly continuous if there is a positive real number such that function values over any function domain interval of the size are as close to each other as we want. In other words, for a uniformly continuous real function of real numbers, if we want function value differences to be less than any positive real number , then there is a positive real number such that at any and in any function interval of the size .

Related lectures (897)

Derivability and DifferentiabilityMATH-101(g): Analysis I

Covers derivability, differentiability, rules of differentiation, and the relationship between differentiability and continuity.

Derivability and ContinuityMATH-101(g): Analysis I

Explores derivability, continuity, and composite functions with illustrative examples.

Properties of Continuous FunctionsMATH-101(g): Analysis I

Explores the continuity of elementary functions and the properties of continuous functions on closed intervals.

Continuous Functions and DerivativesMATH-101(e): Analysis I

Covers the theorem of the intermediate value, derivatives, and their geometric interpretation.

Derivability and Maximum ValuesMATH-101(g): Analysis I

Covers the theorem of intermediate values and finding maximum and minimum values of functions on closed intervals.