Logarithmic Differentiation

    Logarithmic differentiation is a procedure that uses the chain rule and implicit differentiation. Basically the idea is to apply an appropriate logarithmic function to both sides of the given equation and then use some properties of logarithms to simplify before using implicit differentiation.

Proposition (Logarithmic Differentiation) Suppose that is a given equation involving both and ; and that exists at Then can be found using the following procedure (called logarithmic differentiation):

    (i) Apply a logarithmic function with the appropriate base to both sides.
    
    (ii) Use properties of logarithms to simplify.

    (iii) Differentiate both sides of the equation with respect to
    
    (iv) If possible, solve the differentiated equation algebraically for
    

Example (Logarithmic Differentiation) Use logarithmic differentiation to find given

    Solution. Using the natural logarithmic function, we have



and applying implicit differentiation we have,





as desired.

Example (Logarithmic Differentiation) Use logarithmic differentiation to find given

    Solution. Using the natural logarithmic function, we have



and applying implicit differentiation we have,





as desired.

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Logarithmic Differentiation
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
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