Logarithmic differentiation

[Lippi86]

f(x) and g(x) are two differentiable functions and F(x) > 0 for all x. Consider h(x)=f(x)^g(x).

a) Show that ln h(x) = g(X) ln f(x).

b) Show that the derivative of ln h(x) with respect to x is g´(x) ln f(x) + g(x) f´(x)/f(x).

c) Show that the derivative of ln h(x) with respect to x is h´(x)/h(x)

d) Show that h´(x) = f(x)^g(x)[g´(x) lnf(x) + g(x)f´(x)/f(x)]

e) Use the above formula to find the derivative of h(x) = X^x. Identify the functions f(x) and
g(x) and their derivatives.

Thannks for all kind of help!

 

[mmm4444bot]

Hi Lippi:

I need to tell you that we have some informal standards, on these boards. (See the post titled, "Read Before Posting".)

Basically, we like to see you make some statements about what you already understand versus what you don't.

Posting your work, so far, on each exercise is even better!

In the absence of specific information from you, we might be wasting our time by typing a bunch of stuff that you already know.

It's much easier to tutor students when we have some idea about why they are stuck because then we know where to being assisting.

So, the better the information about your situation that you provide, the better the responses that you will get back from us.

Okay?

 

[swaroop]

Since the inportant part of the problem is integrating i will explain how to diff log fubctions.
Say f=g^h (say all r functions of variable x
Taking ln on both sides
ln(f)=h*ln(g)
Diff on both sides
(1/f)*df/dx=(dh/dx)*ln(g)+h*((1/g)*dg/dx)
Remeber g'=dg/dx