Logarithmic differentiation

Logarithmic differentiation takes the logarithm of a function before differentiating, turning products, quotients, and variable powers into simpler sums, differences, and products.

Chain rule: A differentiation rule for composite functions: the outer derivative is multiplied by the inner derivative. Differentiating the logarithm introduces a factor of the original function's derivative.
Variable exponent: An expression in which the exponent depends on a variable, such as x raised to a function of x. Taking its logarithm brings the exponent down as a factor.
Derivative: A quantity describing a function's instantaneous rate of change with respect to its input. Logarithmic differentiation is a technique for finding this quantity.
Direct differentiation: Finding a derivative by applying differentiation rules directly to the original expression. This is the alternative when logarithms add complexity rather than simplify the calculation.
Product rule: A rule stating that the derivative of a product is the first factor times the second derivative plus the second factor times the first derivative. Taking logarithms converts products into sums, avoiding repeated applications of this rule.
Power rule: A differentiation rule giving the derivative of x to a constant power as that power times x to one lower power. Logarithmic differentiation extends the familiar rule to powers whose exponents vary.
Exponential function: A function of the form a raised to x; the natural exponential eˣ is its standard calculus case. Exponentials and logarithms are inverse functions, linking variable powers to logarithmic derivatives.
Power series: An infinite sum of powers of a variable that can represent a function within a convergence interval. Series expansion can provide derivatives when a logarithmic transformation is unsuitable.
Quotient rule: A differentiation rule for a quotient, formed from the denominator times the numerator derivative minus the numerator times the denominator derivative, divided by the denominator squared. Logarithms turn quotients into differences, often making this rule unnecessary.
Generalized power rule: A differentiation rule for a variable raised to a constant power, incorporating the chain rule for its base. It handles variable bases, while logarithmic differentiation also handles variable exponents.

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Differentiation rulesRelated: It combines logarithm identities with product and chain rules.