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)}


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Quote on Mathematics

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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.    

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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