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  2. Logarithmic derivative - Wikipedia

    en.wikipedia.org/wiki/Logarithmic_derivative

    The logarithmic derivative is then / and one can draw the general conclusion that for f meromorphic, the singularities of the logarithmic derivative of f are all simple poles, with residue n from a zero of order n, residue −n from a pole of order n. See argument principle. This information is often exploited in contour integration.

  3. List of logarithmic identities - Wikipedia

    en.wikipedia.org/wiki/List_of_logarithmic_identities

    The multiple valued version of log(z) is a set, but it is easier to write it without braces and using it in formulas follows obvious rules. log(z) is the set of complex numbers v which satisfy e v = z; arg(z) is the set of possible values of the arg function applied to z. When k is any integer:

  4. Differentiation rules - Wikipedia

    en.wikipedia.org/wiki/Differentiation_rules

    Logarithmic differentiation is a technique which uses logarithms and its differentiation rules to simplify certain expressions before actually applying the derivative. [ citation needed ] Logarithms can be used to remove exponents, convert products into sums, and convert division into subtraction — each of which may lead to a simplified ...

  5. Logarithmic differentiation - Wikipedia

    en.wikipedia.org/wiki/Logarithmic_differentiation

    It can also be useful when applied to functions raised to the power of variables or functions. Logarithmic differentiation relies on the chain rule as well as properties of logarithms (in particular, the natural logarithm, or the logarithm to the base e) to transform products into sums and divisions into subtractions.

  6. Logarithm - Wikipedia

    en.wikipedia.org/wiki/Logarithm

    Moreover, as the derivative of f(x) evaluates to ln(b) b x by the properties of the exponential function, the chain rule implies that the derivative of log b x is given by [35] [37] ⁡ = ⁡. That is, the slope of the tangent touching the graph of the base- b logarithm at the point ( x , log b ( x )) equals 1/( x ln( b )) .

  7. Natural logarithm - Wikipedia

    en.wikipedia.org/wiki/Natural_logarithm

    The natural logarithm of a positive real number may also be defined as the derivative of the function = at = (assuming has been previously defined without using the natural logarithm). Using the definition of the derivative as a limit, this definition may be written as ln ⁡ ( a ) = lim x → 0 a x − 1 x . {\displaystyle \ln(a)=\lim _{x\to 0 ...

  8. Liouville's theorem (differential algebra) - Wikipedia

    en.wikipedia.org/wiki/Liouville's_theorem...

    However, is not necessarily equipped with a unique logarithm; one might adjoin many "logarithm-like" extensions to . Similarly, an exponential extension is a simple transcendental extension that satisfies D t t = D s for some s ∈ F . {\displaystyle {\frac {Dt}{t}}=Ds\quad {\text{ for some }}s\in F.}

  9. Power rule - Wikipedia

    en.wikipedia.org/wiki/Power_rule

    Solving for , = = = = = Thus, the power rule applies for rational exponents of the form /, where is a nonzero natural number. This can be generalized to rational exponents of the form p / q {\displaystyle p/q} by applying the power rule for integer exponents using the chain rule, as shown in the next step.