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#### Petrus

##### Well-known member

- Feb 21, 2013

- 739

Hello MHB,

I got one exempel that I dont get same result as my book.

Exempel: If \(\displaystyle z=f(x,y)\) has continuos second-order partial derivates and \(\displaystyle x=r^2+s^2\) and \(\displaystyle y=2rs\) find \(\displaystyle \frac{d^2z}{dr^2}\)

So what I did before checking soulotion:

\(\displaystyle \frac{d^2z}{dr^2}=\frac{dz}{dr} \frac{d}{dr}\)

So I start with solving \(\displaystyle \frac{dz}{dr}=\frac{dz}{dx}\frac{dx}{dr}+\frac{dz}{dy}\frac{dy}{dr} = \frac{dz}{dx}(2r)+\frac{dz}{dy}(2s)\)

so now we got \(\displaystyle \frac{d}{dr}(\frac{dz}{dx}(2r)+\frac{dz}{dy}(2s))\) and that is equal to \(\displaystyle 2r\frac{d}{dr}\frac{dz}{dx}+2s\frac{d}{dr}\frac{dz}{dy}\)

my book soloution:

\(\displaystyle \frac{d^2z}{dr^2}= \frac{d}{dr}(\frac{dz}{dx}(2r)+\frac{dz}{dy}(2s))\) and they equal that to \(\displaystyle 2\frac{dz}{dx}+2r\frac{d}{dr}(\frac{dz}{dx})+2s \frac{d}{dr}(\frac{dz}{dy})\)

my question is where do they get that extra \(\displaystyle 2\frac{dz}{dx}\) (I suspect it was a accident) and my last question why do they got parantes on \(\displaystyle (\frac{dz}{dx})\) and \(\displaystyle (\frac{dz}{dy})\)

then they solve \(\displaystyle \frac{d}{dr} (\frac{dz}{dx})\) and \(\displaystyle \frac{d}{dr}(\frac{dz}{dx})\)

why do they do that and how do they do it? unfortently I have to run so I cant post fully soloution, will post it later.

Regards,

I got one exempel that I dont get same result as my book.

Exempel: If \(\displaystyle z=f(x,y)\) has continuos second-order partial derivates and \(\displaystyle x=r^2+s^2\) and \(\displaystyle y=2rs\) find \(\displaystyle \frac{d^2z}{dr^2}\)

So what I did before checking soulotion:

\(\displaystyle \frac{d^2z}{dr^2}=\frac{dz}{dr} \frac{d}{dr}\)

So I start with solving \(\displaystyle \frac{dz}{dr}=\frac{dz}{dx}\frac{dx}{dr}+\frac{dz}{dy}\frac{dy}{dr} = \frac{dz}{dx}(2r)+\frac{dz}{dy}(2s)\)

so now we got \(\displaystyle \frac{d}{dr}(\frac{dz}{dx}(2r)+\frac{dz}{dy}(2s))\) and that is equal to \(\displaystyle 2r\frac{d}{dr}\frac{dz}{dx}+2s\frac{d}{dr}\frac{dz}{dy}\)

my book soloution:

\(\displaystyle \frac{d^2z}{dr^2}= \frac{d}{dr}(\frac{dz}{dx}(2r)+\frac{dz}{dy}(2s))\) and they equal that to \(\displaystyle 2\frac{dz}{dx}+2r\frac{d}{dr}(\frac{dz}{dx})+2s \frac{d}{dr}(\frac{dz}{dy})\)

my question is where do they get that extra \(\displaystyle 2\frac{dz}{dx}\) (I suspect it was a accident) and my last question why do they got parantes on \(\displaystyle (\frac{dz}{dx})\) and \(\displaystyle (\frac{dz}{dy})\)

then they solve \(\displaystyle \frac{d}{dr} (\frac{dz}{dx})\) and \(\displaystyle \frac{d}{dr}(\frac{dz}{dx})\)

why do they do that and how do they do it? unfortently I have to run so I cant post fully soloution, will post it later.

Regards,

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