# Thread: partial derivatives in finance

1. ## partial derivatives in finance

Hello. I am studying for my exam and just came across a problem that states I must take a partial derivative of a utility function. I am kind of confused since it's been a while since I've done this. The answer in the book appears to be missing a decimal point. But I also need a refresher on partial derivatives it seems . Please note that w1 and w2 are just terms.
The utility function is $w1^{0.6}w2^{0.4}$ and we must figure out it's derivative as in $\frac {dW2}{dW1}$

i got dw2/dw1 = $\frac {0.6w1^{-0.4}w2^{0.4}}{0.4w2^{-0.6}w1^{0.6}}$

from here i am unsure as to why this is equal to $\frac {1.5w2}{w1}$

any help is appreciated

thank you
chris

2. ## Re: partial derivatives in finance

Originally Posted by ffezz
Hello. I am studying for my exam and just came across a problem that states I must take a partial derivative of a utility function. I am kind of confused since it's been a while since I've done this. The answer in the book appears to be missing a decimal point. But I also need a refresher on partial derivatives it seems . Please note that w1 and w2 are just terms.
The utility function is $w1^{0.6}w2^{0.4}$ and we must figure out it's derivative as in $\frac {dW2}{dW1}$

1. Are the 1 and 2 subscripts? If so, you can type them like this: [TEX]w_{1}, w_{2},[/TEX] which produces $w_{1}, w_{2}.$

2. Also, math is case-sensitive. $W_{1}\not=w_{1},$ in general. Are they the same variables for you?

3. Without a starting equation, you cannot compute a derivative. That is, unless you know that $y=x^{2},$ you cannot compute that $y'=2x.$ Let's say that your utility function is denoted by $u.$ Are you told that $u=w_{1}^{0.6}w_{2}^{0.4}?$ If so, then precisely what derivative are you asked to compute?

i got dw2/dw1 = $\frac {0.6w1^{-0.4}w2^{0.4}}{0.4w2^{-0.6}w1^{0.6}}$

from here i am unsure as to why this is equal to $\frac {1.5w2}{w1}$
Well,

$\frac{0.6w_{1}^{-0.4}w_{2}^{0.4}}{0.4w_{2}^{-0.6}w_{1}^{0.6}}=\frac{1.5w_{2}^{0.6}w_{2}^{0.4}}{ w_{1}^{0.4}w_{1}^{0.6}}=\frac{1.5w_{2}}{w_{1}}.$

That's arithmetic and exponentiation laws.

3. ## Re: partial derivatives in finance

Thank you very much Ackbeet All your assumptions are correct. I will try to be more careful with case and equation setting in the future.
I simply didn't see the re-arranging. It makes perfect sense now that you included the intermediate step.