# chain rule

Let $f,g$ be differentiable^{},
real-valued functions such that $g$ is defined on an open set
$I\subseteq \mathbb{R}$, and $f$ is defined on $g(I)$.
Then the derivative^{} of the composition^{} $f\circ g$ is given by
the *chain rule ^{}*, which asserts that

$${(f\circ g)}^{\prime}(x)=({f}^{\prime}\circ g)(x){g}^{\prime}(x),x\in I.$$ |

The chain rule has a particularly suggestive appearance in terms of the Leibniz formalism. Suppose that $z$ depends differentiably on $y$, and that $y$ in turn depends differentiably on $x$. Then we have

$$\frac{dz}{dx}=\frac{dz}{dy}\frac{dy}{dx}.$$ |

The apparent cancellation of the $dy$ term is at best a formal
mnemonic, and does not constitute a rigorous proof of this result.
Rather, the Leibniz format is well suited to the interpretation^{} of the
chain rule in terms of related rates. To wit:

The instantaneous rate of change of $z$ relative to $x$ is equal to the rate of change of $z$ relative to $y$ times the rate of change of $y$ relative to $x$.

Title | chain rule |
---|---|

Canonical name | ChainRule |

Date of creation | 2013-03-22 12:26:43 |

Last modified on | 2013-03-22 12:26:43 |

Owner | matte (1858) |

Last modified by | matte (1858) |

Numerical id | 12 |

Author | matte (1858) |

Entry type | Theorem |

Classification | msc 26A06 |

Related topic | Derivative |

Related topic | ChainRuleSeveralVariables |

Related topic | ExampleOnSolvingAFunctionalEquation |

Related topic | GudermannianFunction |