logarithmic proof of product rule

Following is a proof of the product ruleMathworldPlanetmath using the natural logarithmMathworldPlanetmathPlanetmath, the chain ruleMathworldPlanetmath, and implicit differentiationMathworldPlanetmath. Note that circular reasoning does not occur, as each of the concepts used can be proven independently of the product rule.


Let f and g be differentiable functions and y=f(x)g(x). Then lny=ln(f(x)g(x))=lnf(x)+lng(x). Thus, 1ydydx=f(x)f(x)+g(x)g(x). Therefore,


Once students are familiar with the natural logarithm, the chain rule, and implicit differentiation, they typically have no problem following this proof of the product rule. Actually, with some prompting, they can produce a proof of the product rule to this one. This exercise is a great way for students to review many concepts from calculusMathworldPlanetmath.

Title logarithmic proof of product rule
Canonical name LogarithmicProofOfProductRule
Date of creation 2013-03-22 16:18:48
Last modified on 2013-03-22 16:18:48
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 6
Author Wkbj79 (1863)
Entry type Proof
Classification msc 26A06
Classification msc 97D40
Related topic LogarithmicProofOfQuotientRule