Logarithmic derivatives

Log laws:

log (ab) = log a + log b

log ( ) = log a – log b

log aⁿ = n log a

Logarithmic derivatives

The derivative of the natural log is:

The derivative of the log to the base ‘a’ is:

Logarithmic differentiation:

Consider y= f(x) is a nasty combination of products. In this case we have to follow the following trick:

• Take In on both sides: In y=  In(f(x))
• Use the laws of logs to simplify the right hand side as much as possible.
• Take the derivative (with respect to x) of both sides. Use chain rule on the left hand side.

  = (RHS)’ ,( where y’=dy/dx)

• Solve for y’ by multiplying both sides by the original function.

  y’= f(x) . (RHS)’

Examples:

1. Find the derivative of y=  In ( x⁴sin²x)

Consider y= In (x⁴ sin²x)

 y = In x⁴ + In ( sin²x) ,( apply product rule of log)

 y= 4 In x + 2In (sinx) , (apply power rule of log)

2. Find the derivative of y =

This is a very nasty function. It is the product of three functions, so if we take the derivative directly, we have to use the product rule twice. Also, two of these functions and (x²+1)¹⁰ have functions inside other functions, so we need to use the chain rule for those. The natural log will convert the product of functions into a sum of functions, and it will eliminate powers/exponents.

Consider the given function y=

Taking log on both sides we get,

In y = In (√x) +In (e^x2) + In (x²+1)¹⁰  ,( by product rule of log)

In y = ½ In (x) + x² (In e) + 10 In ( x²+1),( by power rule)

In y = ½  In(x) + x² + 10 In ( x² + 1) ,( because In e = 1)