Logarithmic differentiation, Mathematics

Assignment Help:

Logarithmic Differentiation : There is one final topic to discuss in this section. Taking derivatives of some complicated functions can be simplified by using logarithms.  It is called logarithmic differentiation.

It's easiest to illustrate how this works in an example.

Example Differentiate the function.

1985_Logarithmic Differentiation.png

Solution: Differentiating this function could be completed with a product rule & quotient rule. Though, that would be a fairly messy procedure. We can make simpler things somewhat by taking logarithms of both sides.

1147_Logarithmic Differentiation1.png

Certainly, it isn't really simpler.  What we have to do is utilize the properties of logarithms to expand the right side as follows.

489_Logarithmic Differentiation3.png

It doesn't look all that simple.  Though, the differentiation procedure will be simpler.  What we have to do at this point is differentiate both of the sides w.r.t x.  Note as well that it is really implicit differentiation.

y′ /y= 5x4 / x5 = -10/(1-10x) - ((1/2)   (x2+2)(-1/2)(2x)/(x2+2)(1/2)

y′ /y= 5/ x +10/(1-10x) - x/ (x2+2)

To finish the problem all that we have to do is multiply both sides through y and the plug in for y as we do know what that is.

y′ = y ( 5/x +   (10/1 -10 x)    - x/ x2 + 2))    

1700_Logarithmic Differentiation4.png

Based upon the person, doing this would perhaps be slightly easier than doing both the quotient & product rule. The answer is approximately definitely simpler than what we would have gotten by using the product & quotient rule.

We can also utilize logarithmic differentiation to differentiation functions in the form.

                                    y = ( f ( x ))g ( x )


Related Discussions:- Logarithmic differentiation

Differentiate y = x x using implicit differentiation, Differentiate y = x ...

Differentiate y = x x Solution : We've illustrated two functions similar to this at this point. d ( x n ) /dx = nx n -1                                 d (a x ) /dx= a

Application of statistics-forecasting, Forecasting Statistics is very ...

Forecasting Statistics is very significant for business managers while predicting the future of a business for illustration if a given business situation includes a independen

#algebra, what is the answer of 6_5x9_4x3(1_2)

what is the answer of 6_5x9_4x3(1_2)

Calculate the area of circle, Calculate the area of CIRCLE ? A circle i...

Calculate the area of CIRCLE ? A circle is a set of all points that are at a given distance from a center point. The diameter (d) of a circle is the length of a line that goes

Determine the probability - mean and standard deviation, The scores of stud...

The scores of students taking the ACT college entrance examination are normally distributed with a mean m = 20.1 and a standard deviation s = 5.8. A single student is selected a

Find the maximum expected holdings, Problem: A person has 3 units of mo...

Problem: A person has 3 units of money available for investment in a business opportunity that matures in 1 year. The opportunity is risky in that the return is either double o

Example to understand division means, My nephew had been introduced to divi...

My nephew had been introduced to division by his teacher Ms. Santosh, in Class 3. He, and several of his friends who had been taught by her, appeared to be quite comfortable with t

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd