Quotient rule | EPFL Graph Search

Are you an EPFL student looking for a semester project?

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let , where both f and g are differentiable and The quotient rule states that the derivative of h(x) is It is provable in many ways by using other derivative rules. Given , let , then using the quotient rule: The quotient rule can be used to find the derivative of as follows: Reciprocal rule The reciprocal rule is a special case of the quotient rule in which the numerator . Applying the quotient rule gives Note that utilizing the chain rule yields the same result. Let Applying the definition of the derivative and properties of limits gives the following proof, with the term added and subtracted to allow splitting and factoring in subsequent steps without affecting the value:The limit evaluation is justified by the differentiability of , implying continuity, which can be expressed as . Let so that The product rule then gives Solving for and substituting back for gives: Let Then the product rule gives To evaluate the derivative in the second term, apply the reciprocal rule, or the power rule along with the chain rule: Substituting the result into the expression gives Let Taking the absolute value and natural logarithm of both sides of the equation gives Applying properties of the absolute value and logarithms, Taking the logarithmic derivative of both sides, Solving for and substituting back for gives: Note: Taking the absolute value of the functions is necessary for the logarithmic differentiation of functions that may have negative values, as logarithms are only real-valued for positive arguments. This works because , which justifies taking the absolute value of the functions for logarithmic differentiation. Implicit differentiation can be used to compute the nth derivative of a quotient (partially in terms of its first n − 1 derivatives).

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related concepts (10)

Related courses (27)

Related lectures (222)

Related MOOCs (4)

Neuro Robotics

At the same time, several different tutorials on available data and data tools, such as those from the Allen Institute for Brain Science, provide you with in-depth knowledge on brain atlases, gene exp

Neurorobotics

The MOOC on Neuro-robotics focuses on teaching advanced learners to design and construct a virtual robot and test its performance in a simulation using the HBP robotics platform. Learners will learn t

Neurorobotics

EE-559: Deep learning

This course explores how to design reliable discriminative and generative neural networks, the ethics of data acquisition and model deployment, as well as modern multi-modal models.

MATH-106(en): Analysis II (English)

The course studies fundamental concepts of analysis and the calculus of functions of several variables.

CS-101: Advanced information, computation, communication I

Discrete mathematics is a discipline with applications to almost all areas of study. It provides a set of indispensable tools to computer science in particular. This course reviews (familiar) topics a

Quotient rule

Product rule

In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as or in Leibniz's notation as The rule may be extended or generalized to products of three or more functions, to a rule for higher-order derivatives of a product, and to other contexts. Discovery of this rule is credited to Gottfried Leibniz, who demonstrated it using differentials. (However, J. M.

Logarithmic derivative

In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formula where is the derivative of f. Intuitively, this is the infinitesimal relative change in f; that is, the infinitesimal absolute change in f, namely scaled by the current value of f. When f is a function f(x) of a real variable x, and takes real, strictly positive values, this is equal to the derivative of ln(f), or the natural logarithm of f.

Mediatrix and Bisector

Explores the mediatrix and bisector properties in right-angled triangles through proofs and constructions.

Oja's Rule MOOC: Neuro Robotics

Covers Oja's rule in Neurorobotics, focusing on learning eigenvectors and eigenvalues for capturing maximal variance.

Limit Properties: Exponential and Logarithmic Functions MATH-101(e): Analysis I

Covers the properties of limits related to exponential and logarithmic functions.