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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:
The logarithmic derivative is another way of stating the rule for differentiating the logarithm of a function (using the chain rule): () ′ = ′, wherever is positive. Logarithmic differentiation is a technique which uses logarithms and its differentiation rules to simplify certain expressions before actually applying the derivative.
In summary, both derivatives and logarithms have a product rule, a reciprocal rule, a quotient rule, and a power rule (compare the list of logarithmic identities); each pair of rules is related through the logarithmic derivative.
How to establish this derivative of the natural logarithm depends on how it is defined firsthand. If the natural logarithm is defined as the integral ln x = ∫ 1 x 1 t d t , {\displaystyle \ln x=\int _{1}^{x}{\frac {1}{t}}\,dt,} then the derivative immediately follows from the first part of the fundamental theorem of calculus .
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 )) .
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] () ′ = ′ ′ = () ′.
In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices.It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities.
The derivative of the function given by () = + + is ′ = + () () + = + (). Here the second term was computed using the chain rule and the third term using the product rule. The known derivatives of the elementary functions , , (), (), and =, as well as the constant , were also used.