What is Logarithmic Differentiation?

Logarithmic differentiation differentiates functions which are quotients, products or powers of other functions. This differentiation method is used when direct differentiation becomes complicated. Logarithmic differentiation is useful for functions like $y = {f\left(x\right)}^{g\left(x\right)}$ . Formula for log differentiation of a function is $\frac{d}{dx} \left( x^x \right) = x^x ( 1+ \ln (x\left) \right)$

Logarithmic differentiation is not possible in the following scenarios that CBSE board must be aware of:

1. When you have a function of $y = {f\left(x\right)}^{g\left(x\right)}$ , direct differentiation is not possible using standard rules such as power rule or exponential rule.
2. Whenever there is a product or quotient of multiple functions like y=u(x)⋅v(x)⋅w(x), logarithmic differentiation is used. By taking logarithms, products are converted into sums and quotients into differences. This makes differentiation simpler.
3. Logarithmic differentiation is used when exponent itself is a function of x such as $y = { ( 1+x^2 ) }^{ \sqrt{x} }$
4. Logarithm differentiation simplifies exponentiation. This allows effective use of the chain rule.
5. Direct differentiation is not easy to use when a function is a complex combination of quotients, products and exponents.
6. Do note that logarithmic differentiation should not be used for simple polynomials or basic exponential functions. In these simple functions, standard differentiations are sufficient. Another case, in which logarithmic differentiation should not be used for those functions which do not involve products, variable exponents and quotients. 

The NEET exam or JEE Main exam will not ask you direct definition-based questions. Instead, questions around the applications of these properties. Let us take a look at the important properties of logarithm differentiation:

• The logarithm of product of two functions is equal to sum of logarithms of individual function. This is a useful property of logarithm differentiation because it is easier to differentiate a sum rather than directly differentiating a product. Logarithmic differentiation simplifies the differentiation process by converting products into sum. 
• Logarithm of a quotient of two functions is equal to difference of the logarithms of numerator and denominator. This simplifies process of differentiation by converting complex quotients into manageable differences.
• The logarithm of a function raised to a power is equal to power multiplied by logarithm of function. This is a property useful for dealing with variable exponents since it transforms exponentiation into a multiplication that is much easier to be differentiated.
• Logarithmic differentiation works more effectively for those functions where both base and exponents are functions of x. 
• The derivative of ln y w.r.t. x is $\frac{1}{y} \cdot \frac{dy}{dx}$ when we take the natural logarithm of both sides of an equation that involve a function y.

Let us understand steps refor logarithmic differentiation since questions based on this will be asked in IIT JAM exam and IISER entrance exam.

1. Consider the function y = f(x) that needs to be differentiated
2. Now, take natural logarithm on both the sides ln(y) = ln(f(x))
3. Differentiate both sides of the above equation w.r.t. x. Please note that y is the function of x which means that you will have to use chain rule on the left side.
4. Derivative of ln(y) concerning x is $\frac{1}{y} \cdot \frac{dy}{dx}$
5. Derivative of ln (f(x)) concerning x is $\frac{ f^{\prime } \left(x\right) }{ f \left(x\right) }$
6. Once we have differentiated the function, solve for $\frac{dy}{dx}$ by multiplying y on both sides of the equation.