Some help with this differentiation question thanks

[mikecrush]

some help with this differentiation question thanks
Question :
Find the indicated partial derivative . frss , frst

f(r,s,t) = r ln (rs^2t^3)

differentiating with respect to r gave

s^2t^3
1* ----------
rs^2 t^3

but this is not correct something is missing according to the answer sheet

after getting the fr correct then only i can proceed to frss and frst

thanks in advance

 

[n!kofeyn]

Remind me of the order of differentiation. For example, does f_rst mean take the derivative with respect to r then s then t, or does it mean take the derivative with respect to t then s then r? I ask because I've seen both ways used, and it sometimes changes from book to book. So double check in your book.

Also, if you're taking that derivative with respect to r, then you need to use the product rule.

 

[mikecrush]

Remind me of the order of differentiation. For example, does f_rst mean take the derivative with respect to r then s then t, or does it mean take the derivative with respect to t then s then r? I ask because I've seen both ways used, and it sometimes changes from book to book. So double check in your book.

Also, if you're taking that derivative with respect to r, then you need to use the product rule.

i have double checked my notes and i see that for continuous functions either way is applicable . but for this i believe starts from fr to frs then frss .

the final answer for frss should be -2s^-2

thanks

 

[n!kofeyn]

i have double checked my notes and i see that for continuous functions either way is applicable . but for this i believe starts from fr to frs then frss .

the final answer for frss should be -2s^-2

thanks

That's not true, and you have to be careful. Either way is applicable only if the partial derivatives exist and are continuous themselves (not just the original function) on the domain of your original function. In other words, you can only switch the order of partial differentiation if (in this case) all the third partial derivatives exist and are continuous. For this f, the natural log is only defined for values greater than 0, and so in this case, you can switch the order of integration because the only points at which your partial derivatives have a possibility of failing to exist are going to be where s, r, or t are zero.

Did you use the product rule like I suggested, and then take the other partial derivatives?

 

[mikecrush]

That's not true, and you have to be careful. Either way is applicable only if the partial derivatives exist and are continuous themselves (not just the original function) on the domain of your original function. In other words, you can only switch the order of partial differentiation if (in this case) all the third partial derivatives exist and are continuous. For this f, the natural log is only defined for values greater than 0, and so in this case, you can switch the order of integration because the only points at which your partial derivatives have a possibility of failing to exist are going to be where s, r, or t are zero.

Did you use the product rule like I suggested, and then take the other partial derivatives?

thanks for the heads up

I did the product rule part and natural log differentiates differently from normal differentiation is it ?

from the question above :
f(r,s,t) = r ln (rs^2t^3)

i differentiate fr first

i get

r(s^2t^3)
------------ + lnrs^2t^3
rs^2t^3

is this right ?

i am doing this from left to right .. fr then frs then frst , but can i do it from right to left too ?

thanks

 

[n!kofeyn]

Yes, that's correct. Left to right is fine. Both of the partial derivatives you need contain the f_r, so you've done 1/3 of both problems already. It's just a messy problem either way.

 

[mikecrush]

thank you very much n!kofeyn , really appreciate your help