Definition and mrthod of differentiation :-
Logarithmic differentiation is a very useful method to differentiate some complicated functions which can’t be easily differentiated using the common techniques like the chain rule. It can also be employed for the functions that involve many terms that need the application of the product rule or the quotient rule multiple times to be differentiated.
Follow the following steps to find the differentiation of a logarithmic function:
- Take the natural logarithm of the function to be differentiated.
- Use the properties of logarithmic functions to distribute the terms that were initially accumulated together in the original function and were tough to differentiate.
- Differentiate the resulting equation.
- Multiply the equation by the function itself to get the derivative.
Example 1 :-Compute the derivative of the function
Using the properties of the logarithms –
Differentiate with respect to x, and use the chain rule on the right hand side –
Multiplying by y on both sides, and substituting the value of y, we get –
This should give you a pretty good idea about how to apply this method of differentiation to any problem you encounter on the logarithmic functions.
Example 2 :-
substituting the value of y from (i),we get,