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

representative value or an extreme value, A population forms a normal dist...

A population forms a normal distribution with a mean of μ=80 and a standard deviation of o=15. For every samples, compute the z-score for the sample mean and determine whether the

Matrices, find inverse of [1 2 3 2 4 5 3 5 6]

find inverse of [1 2 3 2 4 5 3 5 6]

Knowing your learner, Here, we have tried to present some of the different ...

Here, we have tried to present some of the different thinking and learning processes of preschool and primary school children, in the context of mathematics learning. We have speci

Derive expressions for the mean and variance, On each day t of n days, N cu...

On each day t of n days, N customers of a supermarket were sampled and the number Xt expressing dissatisfaction was recorded. The results suggested that there were good and bad day

Probablity, probability as that of flipping a coin eight times and getting ...

probability as that of flipping a coin eight times and getting all the times the same side of the coin.)

What is exponential functions, What is Exponential Functions ? Exponent La...

What is Exponential Functions ? Exponent Laws Review: A) Ax / Ay = A(x + y) B) Ax / Ay = A(x - y) C) (ABC)x = AxBxCx D) ((Ax)y)z = Axyz E) (A/B)x = Ax /Bx Definition

Example of a function - inflection point, 1. (a) Give an example of a funct...

1. (a) Give an example of a function, f(x), that has an inflection point at (1, 4). (b) Give an example of a function, g(x), that has a local maximum at ( -3, 3) and a local min

Dot product - vector, Dot Product- Vector The other topic for discu...

Dot Product- Vector The other topic for discussion is that of the dot product.  Let us jump right into the definition of dot product. There is given that the two vectors a

Explain prime numbers vs. composite numbers, Normal 0 false f...

Normal 0 false false false EN-IN X-NONE X-NONE MicrosoftInternetExplorer4

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