Page 21 - mathematics

Logarithmic Differentiation Case (1)

* Logarithmic differentiation is a procedure in which Differentiating complicated product and quotient functions:
logarithms are used to convert the task of differentiating
products and quotients for that differentiating sums and y f gh ln y  ln f gh 
differences. It is especially valuable as a means for handling lm lm
complicated product or quotient functions and exponential
functions where variables appear in both the base and the ln y  ln f  ln g  ln h  ln l  ln m
exponent.
Now: differentiating simple sums and differences.
Main Steps in logarithmic Differentiation:

(i) Take logarithms of both sides of a given equation
(ii) Use the laws of Logarithms to simplify.

(iii) Differentiate implicitly with respect to x.

(iv) Solve the resulting equation for y'.

Example Example

dy x 2  1 e5 sin x
dx if y  tan1 x ln x
Find if y  x3 e2x cos(3x)  Find dy
dx
Solution
Solution
We first take the logarithm of both sides:

ln  y   ln x 3 e 2x cos3 (3x ) We first take the logarithm of both sides:

      ln y  ln x3  ln e2x  ln cos3 3x      ln y  ln x 2 1 5  ln e sin x  ln tan1 x  ln ln x 
   ln y  5ln x 2 1  sin x  ln tan1 x  ln ln x 
 ln y   3ln x  2 x  3ln cos3x

Next, differentiate both sides with respect to x:  y 51  2x  cos x  1 1 2  1  1
x 2 1 tan1 x 1 x ln x x
y 3 2 3 1 3 sin 3 x  y
y  x    Next, differentiate both sides with respect to x:
cos 3 x 
 
 3 sin 3x  5 1  cos x 1 1 1 1 
 x cos 3x  x 2 1 tan 1 1 x ln x x 
y   y   2  9    y   y  2x  x 2  



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