Partial derivatives of the natural logs

In summary, a partial derivative of a natural log is a mathematical concept that represents the rate of change of a natural log function with respect to one of its variables while holding all other variables constant. To find a partial derivative, the derivative of the natural log function is taken with respect to the variable in question and the values of the other variables are plugged in and simplified. The difference between a partial derivative and a regular derivative is that a partial derivative only considers the rate of change with respect to one variable, while a regular derivative considers the overall rate of change of a function. Partial derivatives of natural logs are important in various fields of science and there are rules, such as the chain rule and product rule, for finding them accurately.
  • #1
claratanone
2
0
Find the partial derivatives of the following function:

Q=(1/3)logeL+(2/3)logeK

Any help would be much appreciated!

Below is my working out so far:

[tex]\frac{\partial Q}{\partial L}= \frac{\frac{1}{3}}{L}[/tex]

[tex]\frac{\partial Q}{\partial K}= \frac{\frac{2}{3}}{L}[/tex]

Are these correct?
 
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  • #2
I suspect it is just a typo, but you want:

\(\displaystyle \pd{Q}{K}=\frac{\frac{2}{3}}{K}=\frac{2}{3K}\)
 

FAQ: Partial derivatives of the natural logs

What is a partial derivative of a natural log?

A partial derivative of a natural log is a mathematical concept that represents the rate of change of a natural log function with respect to one of its variables while holding all other variables constant.

How do you find a partial derivative of a natural log?

To find a partial derivative of a natural log, you first take the derivative of the natural log function with respect to the variable in question. Then, you plug in the values of the other variables and simplify the expression.

What is the difference between a partial derivative and a regular derivative?

A partial derivative is a derivative that only considers the rate of change with respect to one variable, while holding all other variables constant. A regular derivative, on the other hand, considers the overall rate of change of a function regardless of the variables involved.

Why are partial derivatives of natural logs important?

Partial derivatives of natural logs are important in many fields of science, such as physics, economics, and engineering. They are used to calculate rates of change and optimize functions in multivariable systems.

Are there any rules for finding partial derivatives of natural logs?

Yes, there are rules for finding partial derivatives of natural logs. These include the chain rule, product rule, quotient rule, and power rule. It is important to understand these rules and practice using them to accurately find partial derivatives.

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