Differentiate y = x x using implicit differentiation, Mathematics

Assignment Help:

Differentiate y = x x

Solution : We've illustrated two functions similar to this at this point.

d ( xn ) /dx = nxn -1                                d (a x ) /dx= a x ln a

Neither of these two can imply here since both need either the base or the exponent to be a constant.  In this case the base and the exponent both are variables and thus we have no way to differentiate this function by using only known rules from earlier sections.

However, with logarithmic differentiation we can do this.  First take logarithm of both sides and utilize the logarithm properties to simplify things a little.

ln y = ln x x

ln y = x ln x

Differentiate both sides by using implicit differentiation.

                               y′ / y = ln x + x ( 1 /x)= ln x + 1

As along the first example multiply through y & substitute back in for y.

y′ = y (1 + ln x )

= x x (1 + ln x )

We'll close this section out with a quick recap of all the various ways we've seen of differentiating functions along with exponents.  It is significant to not get all of these confused.


Related Discussions:- Differentiate y = x x using implicit differentiation

Determine the range of given algorithm, The division algorithm says that wh...

The division algorithm says that when a is divided by b, a unique quotient and remainder is obtained. For a fixed integer b where b ≥ 2, consider the function f : Z → Z given by f(

Math Help, 1. Which of the following is greater than 4.3 x 10^9 a. 2.1 x ...

1. Which of the following is greater than 4.3 x 10^9 a. 2.1 x 10^9 b. 3.2 x 10^9 c. 5.3 x 10^9 d. 7.4 x 10^8 2. Which of the following is less than 6.5 x 10^-5 a. 1.4 x 10

Classifying critical points, Classifying critical points : Let's classify ...

Classifying critical points : Let's classify critical points as relative maximums, relative minimums or neither minimums or maximums. Fermat's Theorem told us that all relative

Rank correlation coefficient, Rank Correlation Coefficient Also ident...

Rank Correlation Coefficient Also identified as the spearman rank correlation coefficient, its reasons is to establish whether there is any form of association among two vari

Give introduction to pythagorean theorem, Give Introduction to Pythagorean ...

Give Introduction to Pythagorean Theorem ? The Pythagorean Theorem says that for any right triangle: a 2 + b 2 = c 2 , where c is the hypotenuse, and a and b are the legs. T

Definition of random variables, Q. Definition of Random Variables? Ans...

Q. Definition of Random Variables? Ans. Up to this point, we have been looking at probabilities of different events. Basically, random variables assign numbers to element

Draw a graph model with the adjacency matrix, QUESTION (a) Draw a graph...

QUESTION (a) Draw a graph model with the following adjacency matrix.                         (b) The diagram below shows different cities labelled a to g and z. Also sh

Pre Calc, Find reference angle alpha and thea element of [0 degrees, 1800 d...

Find reference angle alpha and thea element of [0 degrees, 1800 degrees]

Calculate the average, During 2008 the average number of beds required per ...

During 2008 the average number of beds required per day at St Hallam's hospital was 1800.  During the first 50 days of 2008 the average daily requirement for beds was 1830, with a

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