# logarithmic derivative

Given a function $f$, the quantity ${f}^{\prime}/f$ is known as the
*logarithmic derivative ^{}* of $f$. This name comes
from the observation that, on account of the chain
rule

^{},

$${(\mathrm{log}f)}^{\prime}={f}^{\prime}{\mathrm{log}}^{\prime}(f)={f}^{\prime}/f.$$ |

The logarithmic derivative has several basic properties which make it useful in various contexts.

The logarithmic derivative of the product of
functions is the sum of their logarithmic
derivatives. This follows from the product rule^{}:

$$\frac{{(fg)}^{\prime}}{fg}=\frac{f{g}^{\prime}+{f}^{\prime}g}{fg}=\frac{{f}^{\prime}}{f}+\frac{{g}^{\prime}}{g}$$ |

The logarithmic derivative of the quotient of
functions is the difference of their logarithmic
derivatives. This follows from the quotient rule^{}:

$$\frac{{(f/g)}^{\prime}}{f/g}=\frac{{f}^{\prime}g-f{g}^{\prime}}{{g}^{2}}\frac{g}{f}=\frac{{f}^{\prime}}{f}-\frac{{g}^{\prime}}{g}$$ |

The logarithmic derivative of the $p$-th power
of a function is $p$ times the logarithmic
derivative of the function. This follows
from the power rule^{}:

$$\frac{{({f}^{p})}^{\prime}}{{f}^{p}}=\frac{p{f}^{p-1}{f}^{\prime}}{{f}^{p}}=p\frac{{f}^{\prime}}{f}$$ |

The logarithmic derivative of the exponential^{}
of a function equals the derivative^{} of a
function. This follows from the chain rule:

$$\frac{{\left({e}^{f}\right)}^{\prime}}{{e}^{f}}=\frac{{e}^{f}{f}^{\prime}}{{e}^{f}}={f}^{\prime}$$ |

Using these identities, it is rather easy to compute the logarithmic derivatives of expressions which are presented in factored form. For instance, suppose we want to compute the logarithmic derivative of

$${e}^{{x}^{2}}\frac{{(x-2)}^{3}(x-3)}{x-1}.$$ |

Using our identities, we find that its logarithic derivative is

$$2x+\frac{3}{x-2}+\frac{1}{x-3}-\frac{1}{x-1}.$$ |

Title | logarithmic derivative |
---|---|

Canonical name | LogarithmicDerivative |

Date of creation | 2013-03-22 16:47:02 |

Last modified on | 2013-03-22 16:47:02 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 11 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 26B05 |

Classification | msc 46G05 |

Classification | msc 26A24 |

Related topic | ZeroesOfDerivativeOfComplexPolynomial |