differentiable at , then we can write linear approximations: and where the error terms are small with respect to h: that is, also written . Then: The "error terms" consist of items such as and which are easily seen to have magnitude Dividing by and taking the limit gives the result. Quarter squares This proof uses the chain rule and the quarter square function with derivative . We have: and differentiating both sides gives: Multivariable chain rule The product rule can be considered a special case of the chain rule for several variables, applied to the multiplication function : Non-standard analysis Let u and v be continuous functions in x, and let dx, du and dv be infinitesimals within the framework of non-standard analysis, specifically the hyperreal numbers. Using st to denote the standard part function that associates to a finite hyperreal number the real infinitely close to it, this gives This was essentially Leibniz's proof exploiting the transcendental law of homogeneity (in place of the standard part above). Smooth infinitesimal analysis In the context of Lawvere's approach to infinitesimals, let be a nilsquare infinitesimal. Then and , so that since Dividing by then gives or . Logarithmic differentiation Let . Taking the absolute value of each function and the natural log of both sides of the equation, Applying properties of the absolute value and logarithms, Taking the logarithmic derivative of both sides and then solving for : Solving for and substituting back for gives: Note: Taking the absolute value of the functions is necessary to allow logarithmic differentiation of functions that can have negative values, as logarithms are only defined for positive arguments. This works because , which justifies taking the absolute value of the functions for logarithmic differentiation. Generalizations Product of more than two factors The product rule can be generalized to products of more than two factors. For example, for three factors we have For a collection of functions , we have 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 , denoted here , is the derivative of the logarithm of the function. It follows that Using that the logarithm of a product is the sum of the logarithms of the factors, the sum rule for derivatives gives immediately The last above expression of the derivative