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  2. Logarithmic differentiation - Wikipedia

    en.wikipedia.org/wiki/Logarithmic_differentiation

    In calculus, logarithmic differentiation or differentiation by taking logarithms is a method used to differentiate functions by employing the logarithmic derivative of a function f, [1] (⁡) ′ = ′ ′ = (⁡) ′.

  3. 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.

  4. List of logarithmic identities - Wikipedia

    en.wikipedia.org/wiki/List_of_logarithmic_identities

    For example, two numbers can be multiplied just by using a logarithm table and adding. These are often known as logarithmic properties, which are documented in the table below. [2] The first three operations below assume that x = b c and/or y = b d, so that log b (x) = c and log b (y) = d. Derivations also use the log definitions x = b log b (x ...

  5. Differentiation rules - Wikipedia

    en.wikipedia.org/wiki/Differentiation_rules

    The derivative of the function at a point is the slope of the line tangent to the curve at the point. Slope of the constant function is zero, because the tangent line to the constant function is horizontal and its angle is zero. In other words, the value of the constant function, y, will not change as the value of x increases or decreases.

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

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

    Thus, on an intuitive level, the theorem states that the only elementary antiderivatives are the "simple" functions plus a finite number of logarithms of "simple" functions. A proof of Liouville's theorem can be found in section 12.4 of Geddes, et al. [ 4 ] See Lützen's scientific bibliography for a sketch of Liouville's original proof [ 5 ...

  7. Product rule - Wikipedia

    en.wikipedia.org/wiki/Product_rule

    The logarithmic derivative provides a simpler expression of the last form, as well as a direct proof that does not involve any recursion. The logarithmic derivative of a function f, denoted here Logder(f), is the derivative of the logarithm of the function.

  8. Complex logarithm - Wikipedia

    en.wikipedia.org/wiki/Complex_logarithm

    A single branch of the complex logarithm. The hue of the color is used to show the argument of the complex logarithm. The brightness of the color is used to show the modulus of the complex logarithm. The real part of log(z) is the natural logarithm of | z |. Its graph is thus obtained by rotating the graph of ln(x) around the z-axis.

  9. Digamma function - Wikipedia

    en.wikipedia.org/wiki/Digamma_function

    As x goes to infinity, ψ(x) gets arbitrarily close to both ln(x − ⁠ 1 / 2 ⁠) and ln x. Going down from x + 1 to x , ψ decreases by ⁠ 1 / x ⁠ , ln( x − ⁠ 1 / 2 ⁠ ) decreases by ln( x + ⁠ 1 / 2 ⁠ ) / ( x − ⁠ 1 / 2 ⁠ ) , which is more than ⁠ 1 / x ⁠ , and ln x decreases by ln(1 + ⁠ 1 / x ⁠ ) , which is less than ...