Logarithmic Differentiation

In this method logarithmic differentiation we are going to see some examples problems to understand where we have to apply this method.

Example 1:

Differentiate [sin x cos (x²)]/[ x³ + log x ] with respect to x

Solution:

We can differentiate this function using quotient rule, logarithmic-function. When we apply the quotient rule we have to use the product rule in differentiating the numerator. But in the method of logarithmic-differentiation first we have to apply the formulas log(m/n) = log m - log n and log (m n) = log m + log n.

Let y = [sin x cos (x²)]/[ x³ + log x ]

Take log on both sides

log y = log [sin x cos (x²)]/[ x³ + log x ]

log y = log [sin x cos (x²)] - log [ x³ + log x ]

log y = log [sin x ] + log [ cos (x²) ] - log [ x³ + log x ]

(1/y)dy/dx=(1/sin x)cos x-[1/cos (x²)]Sin (x²)(2x)-[1/(x³ + log x )]( 3x² + 1/x )

(1/y)dy/dx=(cos x/sin x)-(2x)sin (x²)/cos (x²)-[1/(x³+log x )]( 3x³ + 1)/x

dy/dx = [cot x  - (2x) tan (x²)  - ( 3x³ + 1)/x(x³ + log x )]y

dy/dx = [cot x  - (2x) tan (x²)  - ( 3x³ + 1)/x(x³ + log x )] x

           [sin x cos (x²)]/[ x³ + log x ]    logarithmic differentiation

---

Example 2:

Differentiate [(x² + 2) (x + √2)]/ [(√(x+4) - (x-7)] with respect to x

Solution:

Let y = [(x² + 2) (x + √2)]/ [(√(x+4) - (x-7)]

log y = log [(x² + 2) (x + √2)]/ [(√(x+4) - (x-7)]

log y = log [(x² + 2) (x + √2)] - log [(√(x+4) - (x-7)]

log y = log (x² + 2) + log  (x + √2) - log [(√(x+4) - (x-7)]

(1/y)dy/dx=[1/(x²+2)]2x+1/(x+√2)-[1/ (√(x+4)-(x-7)]x 1/2√(x+4) - 1

(1/y)dy/dx=[2x/(x²+2)]+1/(x+√2)-{[[1/2√(x+4)] - 1]/ (√(x+4)-(x-7)}

(1/y)dy/dx=[2x/(x²+2)]+1/(x+√2)-{[[1 - 2√(x+4)/2√(x+4)]/ (√(x+4)-(x-7)}

dy/dx=yx [2x/(x²+2)]+1/(x+√2)-{[[1 - 2√(x+4)/2√(x+4)]/ (√(x+4)-(x-7)}

Related Topics

• First Principles 
• Implicit Function
  
• Parametric Function
• Substitution Method
  
• Product Rule
  
• Chain Rule
  
• Quotient Rule
• Rate of Change
• Rolle's theorem
• Lagrange's theorem
• Finding increasing or decreasing interval
  
• Increasing function
  
• Decreasing function
• Monotonic function
• Maximum and minimum
• Examples of maximum and minimum

Quote on Mathematics

“Mathematics, without this we can do nothing in our life. Each and everything around us is math.

Math is not only solving problems and finding solutions and it is also doing many things in our day to day life.  They are: 

It subtracts sadness and adds happiness in our life.    

It divides sorrow and multiplies forgiveness and love.

Some people would not be able accept that the subject Math is easy to understand. That is because; they are unable to realize how the life is complicated. The problems in the subject Math are easier to solve than the problems in our real life. When we people are able to solve all the problems in the complicated life, why can we not solve the simple math problems?

Many people think that the subject math is always complicated and it exists to make things from simple to complicate. But the real existence of the subject math is to make things from complicate to simple.”

Logarithmic Differentiation to First Principles