need help with some partial derivatives: f(x,y,z)=ez+e2yarctan(y/x)

[zwambooo]

This one might be a difficult one:

f(x,y,z)=e^z+e^2yarctan(y/x)

This is what i have calculated

(Partial derivative w.r.t x) dfdx= -(e^2yy)/(y²+x²)
(Partial derivative w.r.t y) dfdy= (e^2yx)/(y²+x²) + 2e^2yarctan(y/x)
(Partial derivative w.r.t z) dfdz= e^z

Here comes the problem
g(s,t)=f(s,u(s,t),v(s,t)), where u(s,t)=st and v(s,t)=s+t
Use the chain rule or invariance of property of differentials to find the first order partial derivatives
(dg/ds and dg/dt). Answer should be expressed in terms of s and t only

can someone please help me?

 

[Ishuda]

This one might be a difficult one:

f(x,y,z)=e^z+e^2yarctan(y/x)

This is what i have calculated

(Partial derivative w.r.t x) dfdx= -(e^2yy)/(y²+x²)
(Partial derivative w.r.t y) dfdy= (e^2yx)/(y²+x²) + 2e^2yarctan(y/x)
(Partial derivative w.r.t z) dfdz= e^z

Here comes the problem
g(s,t)=f(s,u(s,t),v(s,t)), where u(s,t)=st and v(s,t)=s+t
Use the chain rule or invariance of property of differentials to find the first order partial derivatives
(dg/ds and dg/dt). Answer should be expressed in terms of s and t only

can someone please help me?

This is just a simple extension of the one dimensional problem as you have done for the partial derivatives. Consider your problem and what the following gives if u=x, v=y, w=z, and sequentially s=x, then y, then z:
\(\displaystyle \dfrac{\partial f(u,v, w)}{\partial s} = \dfrac{\partial{f(u,v,w)}}{\partial{u}}\, \dfrac{\partial{u}}{\partial{s}}\, +\, \dfrac{\partial{f(u,v,w)}}{\partial{v}}\, \dfrac{\partial{v}}{\partial{s}}\, +\, \dfrac{\partial{f(u,v,w)}}{\partial{w}}\, \dfrac{\partial{w}}{\partial{s}}\)

 

[zwambooo]

@Ishuda

Thank you very much for helping me, but i dont quite get the calculation process of the problem. Do you think you could elaborate some more and maybe show me?

I am thinking that df(u,v,w)/du should be the same as df/dx but instead of x we now have u? But what is the du/ds i.e? or dv/ds for that matter

Im sorry for not comprehending, kinda mind****s me. Could you please show calculation process? Would be appreciated massivly

Edit: first part of the partial derivative is ok, the df(s,u,w)/du. The ones that contain df above the fraction bar, but the ones without im just completly lost on. to be more exact du/ds,dv/ds and dw/ds. i dont even know if its possible to calculate, could you please help me?