chain rule

(noun)

a formula for computing the derivative of the composition of two or more functions.

Related Terms

Examples of chain rule in the following topics:

  • The Chain Rule

    • The chain rule is a formula for computing the derivative of the composition of two or more functions.
    • For example, following the chain rule for $f \circ g(x) = f[g(x)]$ yields:
    • Using the chain rule yields:
    • Use of the chain rule is needed for the complicated calculation.
    • Calculate the derivative of a composition of functions using the chain rule

  • The Chain Rule

    • The chain rule is a formula for computing the derivative of the composition of two or more functions.
    • For example, the chain rule for $f \circ g$ is $\frac {df}{dx} = \frac {df}{dg} \, \frac {dg}{dx}$.
    • The chain rule above is for single variable functions $f(x)$ and $g(x)$.
    • However, the chain rule can be generalized to functions with multiple variables.
    • The chain rule can be used to take derivatives of multivariable functions with respect to a parameter.

  • Differentiation Rules

    • The rules of differentiation can simplify derivatives by eliminating the need for complicated limit calculations.
    • In many cases, complicated limit calculations by direct application of Newton's difference quotient can be avoided by using differentiation rules.
    • Some of the most basic rules are the following.
    • Here the second term was computed using the chain rule and the third using the product rule.
    • The flight of model rockets can be modeled using the product rule.

  • Alkanes

    • Longer chain alkanes are well known, and their names may be found in many reference and text books.
    • Fortunately, common numerical prefixes are used in naming chains of five or more carbon atoms.
    • (iii) These formulas all fit the CnH2n+2 rule.
    • The IUPAC system requires first that we have names for simple unbranched chains, as noted above, and second that we have names for simple alkyl groups that may be attached to the chains.
    • Number the chain consecutively, starting at the end nearest a substituent group.

  • Investment in Operations

    • Investment in information technology has made supply chains faster, cheaper, and more reliable.
    • Supply chain optimization applies processes and tools that ensure optimal operation of a manufacturing and distribution supply chain.
    • Supply chain optimization addresses the general supply chain problem of delivering products to customers at low cost and high profit.
    • Some vendors are applying "best fit" models to this data, to which safety stock rules are applied, while other vendors have started to apply stochastic techniques to the optimization problem.
    • Supply chain optimization applies processes and tools that ensure the optimal operation of a manufacturing and distribution supply chain.

  • Analysis of Molecular Formulas

    • Three useful rules may be listed:
    • The origin of this formula is evident by considering a hydrocarbon made up of a chain of carbon atoms.
    • Thus, a six-carbon chain (n = 6) may be written H-(CH2)6-H, and the total hydrogen count is (2 x 6) + 2 = 14.
    • Consider a hydrocarbon with a molecular structure consisting of a simple chain of four carbon atoms, CH3CH2CH2CH3.
    • The molecular formula is C4H10 (the maximum number of bonded hydrogens by the 2n + 2 rule).

  • Properties of Macromolecules

    • In some cases the entire solid may be amorphous, composed entirely of coiled and tangled macromolecular chains.
    • As noted earlier, HDPE is composed of very long unbranched hydrocarbon chains.
    • As a rule, mechanical properties such as ductility, tensile strength, and hardness rise and eventually level off with increasing chain length.
    • To begin with, cellulose chains easily adopt a stable rod-like conformation.
    • The cis-double bonds in the hydrocarbon chain provide planar segments that stiffen, but do not straighten the chain.

  • Other factors affecting supply chain management

    • In addition to managing the bullwhip effect, supply chain managers must also contend with a variety of factors that pose on-going challenges:
    • This puts added pressure on supply chain managers to continually improve performance.
    • Globalization imposes challenges such as greater geographic dispersion among supply chain members.
    • Government regulations, tariffs, and environmental rules provide challenges as well.

  • Naming Organic Compounds

    • First, it should indicate how the carbon atoms of a given compound are bonded together in a characteristic lattice of chains and rings.
    • The IUPAC nomenclature system is a set of logical rules devised and used by organic chemists to circumvent problems caused by arbitrary nomenclature.
    • Knowing these rules and given a structural formula, one should be able to write a unique name for every distinct compound.
    • A root or base indicating a major chain or ring of carbon atoms found in the molecular structure.
    • A full presentation of the IUPAC Rules is also available.

  • Formulas Using Other Configurational Notations

    • When describing acyclic compounds incorporating two or more chiral centers, many chemists prefer to write zig-zag line formulas for the primary carbon chain.
    • Here, the zig-zag carbon chain lies in a plane and the absolute or relative configurations at the chiral centers are then designated by wedge or hatched bonds to substituent groups.
    • In the commonly used zig-zag drawings substituents may lie on the same side of the carbon chain, a syn orientation, or on opposite sides, an anti orientation.
    • At sites having two substituents, such as carbon #5, the terms refer to the relative orientation of the highest order substituent, as determined by the C.I.P. sequence rules.