When we want to find derivative of some complicated functions, we may have to use more than one applications like product rule, quotient rule and chain rule. It may increase the workings and time consumption. If we use logarithm for such functions, the function can be splitted up into smaller parts using the properties of logaritm rules and the derivative process easier.

This technique is particularly useful for the functions involving products, quotients or powers, as it simplifies the differentiation process.

The following steps will be useful to do logarithmic differentiation of a function y = f(x).

Step 1 :

Take natural logarithm on both sides of y = f(x).

Step 2 :

Apply the properties of logarithm.

Step 3 :

Differentiate both sides with respect to x.

Step 4 :

Solve for ᵈʸ⁄dₓ.

Properties of Logarithms

Product Rule :

ln (ab) = ln a + ln b

Quotient Rule :

ln (ᵃ⁄b) = ln a - ln b

Power Rule :

ln (a^{b}) = bln a

Solved Problems

Find ᵈʸ⁄dₓ in each case.

Problem 1 :

y = x^{x}

Solution :

y = x^{x}

Take natural logarithm on both sides.

ln y = ln x^{x}

ln y = x ⋅ ln x

Find the derivative on both sides with respect to x.

Problem 2 :

Solution :

Find the derivative on both sides with respect to x.

Problem 3 :

Solution :

Find the derivative on both sides with respect to x.

Problem 4 :

Solution :

Find the derivative on both sides with respect to x.

Problem 5 :

x^{m}y^{n} = (x + y)^{m + n}

Solution :

Find the derivative on both sides with respect to x.