Partial derivative

It's when you have a multivariable function but you take the derivative with respect to one variable. You basically treat the other variables as constants.



https://www.youtube.com/watch?v=SbfRDBmyAMI

https://www.khanacademy.org/math/multivariable-calculus/multivariable-derivatives/partial-derivative-and-gradient-articles/a/introduction-to-partial-derivatives

It's when you have a multivariable function but you take the derivative with respect to one variable. You basically treat the other variables as constants.



https://www.youtube.com/watch?v=SbfRDBmyAMI

https://www.khanacademy.org/math/multivariable-calculus/multivariable-derivatives/partial-derivative-and-gradient-articles/a/introduction-to-partial-derivatives

So is second partial derivative basically differentiating twice?

For example (original expression V = RT/P) if the first derivative was (dV/dT)p = R/P then would the second derivative be (dV^2/d^2T)p = 0

This is fine apart from the fact that you have $p$ on the LHS for some reason. It shouldn't be there.

Furthermore, you want to say $\dfrac{\partial V}{\partial T}$ when talking about partial derivatives, instead of $\dfrac{dV}{dT}$. I understand it's easier to just use 'd' on TSR if you don't know how to write the other one, so that's fine, though also note that we have the notation of $f_x, f_y$ for $\dfrac{\partial f}{\partial x}$ and $\dfrac{\partial f}{\partial y}$, respectively. And similalrly, $\dfrac{\partial^2 f}{\partial x^2} = f_{xx}$

Sorry i made a mistake - the equation is RT/p the p on the LHS shows that p is a constant (my lecture notes say this is how you should write it?).

Yes i meant the the other type of d. Oh so in my case would it be (dV/dT)p x (dV/dp)T to get the second derivative? Although the question suggests i should have more than 1 second derivative as the answer??

Er... not sure what your lecture notes are saying about that $p$ as it doesn't seem right.

$V=\dfrac{RT}{p}$

$\displaystyle \Rightarrow \dfrac{\partial V}{\partial T} = \dfrac{R}{p}$

$\displaystyle \Rightarrow \dfrac{\partial }{\partial T} \left( \dfrac{\partial V}{\partial T} \right) = \dfrac{\partial^2 V}{\partial T^2} = 0$

in some notes it has a small subscript letter to represent whats constant outside of the brackets of a partial derivative, im pretty sure its just an obscure piece of notation not many people use

Anyone have any good resources on partial derivatives? - for someone who has never done it before. Thanks

If not please could someone explain the basic concepts?

Could be - I certainly never came across it, only the usual subscript to denote the variable to which we differentiate with respect to.

I'd hate to use their notation for functions of 4 or more variables!

Could be - I certainly never came across it, only the usual subscript to denote the variable to which we differentiate with respect to.

I'd hate to use their notation for functions of 4 or more variables!

in some notes it has a small subscript letter to represent whats constant outside of the brackets of a partial derivative, im pretty sure its just an obscure piece of notation not many people use

Yes its just a small subscript to that tells you what variables are constant. I actually find it quite helpful

Ive attached the original question. Q3 is what ive already calculated. The notation for Q4 confused me and Q5 is what i was asking about the second partial derivative. NOTE: I wrote V instead of Vm just because its easier here.

The only two issues I have with it is that (a) why do we not mention R being a constant as well within the subscript...? and (b) it would get very messy for more variables.



For Q4, just differentiate both sides w.r.t P

For Q5, just take the expressions you found in Q2 and differentiate both w.r.t to the other variable.

Good point! Maybe the person who made the questions made a mistake?

For Q4 I dont understand what im differentiating

Er... not sure what your lecture notes are saying about that $p$ as it doesn't seem right. Posting the full question might help understand what's going on.

$V=\dfrac{RT}{p}$

$\displaystyle \Rightarrow \dfrac{\partial V}{\partial T} = \dfrac{R}{p}$

$\displaystyle \Rightarrow \dfrac{\partial }{\partial T} \left( \dfrac{\partial V}{\partial T} \right) = \dfrac{\partial^2 V}{\partial T^2} = 0$

Hey im getting a bit confused with the notation.

So i have 2 first order derivatives and tried to calculate the 2 second order. Is my notation/answer correct?

Sorry i had to do it on paint so it was easier to see