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

Euilibrium, What is partial market equilibrium

What is partial market equilibrium

Find out ratio, the sides of a right angle triangle are a,a+d,a+2d with a a...

the sides of a right angle triangle are a,a+d,a+2d with a and d both positive.the ratio of a to d  a)1:2 b)1:3 c)3:1 d)5:2 answer is (c) i.e. 3:1 Solution: Applying

Algebra, find the value of A and B if the following polynomials are perfect...

find the value of A and B if the following polynomials are perfect square:

General rule - probability rule, GENERAL RULE A general rule is to sub...

GENERAL RULE A general rule is to subtract the probabilities with an even number of components inside the parentheses and add those with an odd number of components (one or th

Example of operational stages in learning maths, Children of the same age c...

Children of the same age can be at different operational stages, and children of different ages, can be at the same developmental stage." Do you agree with this statement? If so, g

Operation research, advantages of vogel''s approximation method over north ...

advantages of vogel''s approximation method over north west corner method

What is plotting points, What is Plotting Points ? How would you go abo...

What is Plotting Points ? How would you go about drawing the graph of y = x2 ? One way to do it is by plotting points. (Your graphing calculator uses this method.) This is

Differential equation, Verify Liouville''''s formula for y "-y" - y'''' + y...

Verify Liouville''''s formula for y "-y" - y'''' + y = 0 in (0, 1)

Game theory, Game Theory It is used to find out the optimum strategy in...

Game Theory It is used to find out the optimum strategy in a competitive condition,While two or more competitors are engaged in making decisions, this may occupy conflict of in

Ratio, how can i solve it

how can i solve it

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