Easy second order partial derivative

[sainistar]

Hello Experts

I have a simple question.

Given V as the function of Z and Y,
Given Z as the function of R and L, $Z=R+s*L$
Given Y as the function of G and C, $Y=G+s*C$
Assume we also know $\frac{\partial V}{\partial Z}$ and $\frac{\partial^2 V}{\partial Z \partial Y}$
If we want to know $\frac{\partial V}{\partial R}$, it will be equal to $\frac{\partial V}{\partial Z} * \frac{\partial Z}{\partial R}$
Question:
How can we find $\frac{\partial^2 V}{\partial R\partial G}$, by using what is given and known?

Regards
Aman

 

[jvicens]

Hello Aman,

To find \frac{\partial^2 V}{\partial R\partial G}, we can use the chain rule of partial derivatives. The chain rule states that for a function f(x,y) with x and y both functions of another variable t, the partial derivative of f with respect to t can be found by multiplying the partial derivatives of f with respect to x and y by the corresponding derivatives of x and y with respect to t.

In this case, we have V as a function of Z and Y, where Z is a function of R and L and Y is a function of G and C. So we can use the chain rule to find \frac{\partial V}{\partial R} and \frac{\partial V}{\partial G}, and then multiply them together to find \frac{\partial^2 V}{\partial R\partial G}.

Using the chain rule, we have:

\frac{\partial V}{\partial R} = \frac{\partial V}{\partial Z} * \frac{\partial Z}{\partial R}
\frac{\partial V}{\partial G} = \frac{\partial V}{\partial Y} * \frac{\partial Y}{\partial G}

Substituting the given functions for Z and Y, we have:

\frac{\partial V}{\partial R} = \frac{\partial V}{\partial Z} * s
\frac{\partial V}{\partial G} = \frac{\partial V}{\partial Y} * 1

Then, to find \frac{\partial^2 V}{\partial R\partial G}, we multiply these two partial derivatives together:

\frac{\partial^2 V}{\partial R\partial G} = \frac{\partial V}{\partial Z} * s * \frac{\partial V}{\partial Y}

So, using the given information and the chain rule, we can find \frac{\partial^2 V}{\partial R\partial G}.

I hope this helps!