# 242 Derivatives of Logarithmic Functions of y=xlnx-x

• MHB
• karush
In summary, the derivative of $y=\ln{\sqrt{\dfrac{a^2-z^2}{a^2+z^2}}}$ is $\dfrac{-z}{a^2-z^2} - \dfrac{z}{a^2+z^2}$ or simplified as $\dfrac{2a^2 z}{z^4-a^4}$.
karush
Gold Member
MHB
$\tiny{from\, steward\, v8\, 6.4.2}$
find y'
$\quad y= x\ln{x}-x$
so
$\quad y'=(x\ln{x})'-(x)'$
product rule
$\quad (x\ln{x})'=x\cdot\dfrac{1}{x}+\ln{x}\cdot(1)=1+\ln{x}$
and
$\quad (-x)'=-1$
finally
$\quad \ln{x}+1-1=\ln{x}$

well this is an even # with no book answer but I think it is ok
typos maybe
also guess we could of factored out the x but not sure if this would help

$y = x(\ln{x}-1)$

$y' = x \cdot \dfrac{1}{x} + (\ln{x} - 1) \cdot 1 = 1 + \ln{x} - 1 = \ln{x}$

same same

thot I would slip in another one under the table ...

$y=\ln{\sqrt{\dfrac{a^2-z^2}{a^2+z^2}}}$ok calculator says y'=0 but not sure what you do with the 2 variables

Last edited:
karush said:
thot I would slip in another one under the table ...

$y=\ln{\sqrt{\dfrac{a^2-z^2}{a^2+z^2}}}$ok calculator says y'=0 but not sure what you do with the 2 variables

I'm assuming $y$ is a function of $z$ and $a$ is a constant.

using properties of logs ...

$y = \dfrac{1}{2}\left[\ln(a^2-z^2) - \ln(a^2+z^2)\right]$

$\dfrac{dy}{dz} = \dfrac{1}{2}\left(\dfrac{-2z}{a^2-z^2} - \dfrac{2z}{a^2+z^2}\right)$

$\dfrac{dy}{dz} = \dfrac{-z}{a^2-z^2} - \dfrac{z}{a^2+z^2}$

common denominator & combine fractions ...

$\dfrac{dy}{dz} = \dfrac{2a^2 z}{z^4-a^4}$

So y'= 0 at z= 0. Another possibility is that your calculator is assuming that the independent variable is always x and treated both a and z as constants so the whole function is treated as a constant so the derivative is 0.

## 1. What is the definition of a derivative?

A derivative is a mathematical concept that represents the rate of change of a function at a specific point. It is calculated by finding the slope of a tangent line to the function at that point.

## 2. How do I find the derivative of a logarithmic function?

To find the derivative of a logarithmic function, you can use the logarithmic differentiation technique. This involves taking the natural logarithm of both sides of the function, using the properties of logarithms to simplify, and then finding the derivative using the chain rule.

## 3. What is the derivative of y=xlnx-x?

The derivative of y=xlnx-x is y'= ln(x) + 1 - 1/x. This can be found using the logarithmic differentiation technique mentioned above.

## 4. How do I graph the derivative of y=xlnx-x?

To graph the derivative of y=xlnx-x, you can use the graph of the original function and the properties of derivatives. The derivative represents the slope of the tangent line at each point on the original function's graph. By plotting the slope at different points, you can create the graph of the derivative.

## 5. What is the significance of the derivative of y=xlnx-x?

The derivative of y=xlnx-x has several applications in mathematics and science. It can be used to find the maximum and minimum values of a function, as well as to determine the concavity and inflection points of a graph. It is also used in optimization problems and in physics to calculate velocity and acceleration.

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