What is the derivative of #sqrt(xy)#?
It depends on which partial derivative you're interested in. Let's develop some possibilites of derivation.
For our purposes, let's just rewrite it as
Let's find the first partial derivative for
Now, in the same sense, for
Expanding our discussion, as this is a really common production function in Microeconomics (but also used, with some variations, as some Macroeconomics' growth models - even some Nobel prize winners started with a Cobb-Douglas), we can interprete these results.
Considering, for example, a production function where
It is important to underline that the exponents may vary from case to case - or from an economic sector to another. For example, it is possible to have a production function such as
The cool part of it is that when you sum the exponents you get the nature of the company/sector returns: generalizing the function as
If this is one part of a bigger implicit differentiation problem, here's the derivative of this one term with respect to
And the product rule to find: