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

Test, a piece of ribbon measures 2,25 meters . it is cut in half . how long...

a piece of ribbon measures 2,25 meters . it is cut in half . how long is one half of the ribbon

Managment Science, Classify models based on the degree of their abstraction...

Classify models based on the degree of their abstraction, and provide some examples of such models.

Intrgers, how to evaluate the sums

how to evaluate the sums

Why learn mathematics, Here we have considered the following points. 1. ...

Here we have considered the following points. 1. Mathematics is omnipresent, powerful and beautiful. 2. Mathematics is useful in all spheres of life. 3. Mathematics can al

Regression model, Consider the regression model  Y i = a + bX i + u i ,  ...

Consider the regression model  Y i = a + bX i + u i ,  where the  X i   are non-stochastic and the  u i   are independently and identically distributed with  E[u i ] = 0  and  va

Inside function and outside function, "Inside function" and "outside functi...

"Inside function" and "outside function : Generally we don't actually do all the composition stuff in using the Chain Rule. That can get little complexes and actually obscures the

Evaluate limit, Evaluate the given limit. Solution: In this quest...

Evaluate the given limit. Solution: In this question none of the earlier examples can help us. There's no factoring or simplifying to accomplish.  We can't rationalize &

Construction , construct of tangents a circle from an external point when ...

construct of tangents a circle from an external point when its centre is not known

Mathematics for finance, 1. XYZ company’s cost function for the next four m...

1. XYZ company’s cost function for the next four months is C = 600,000 + 8Q a) Find the BEP dollar volume of sales if the selling price is br. 10 / unit b) What woul

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