Let f(x,y,z) = `e^(3x)` sinycosz+ arcsin`(z^(1/3))/(y^2+z^2).` find `f_(yzx).` Thanks! :) ``

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aruv | High School Teacher | (Level 2) Valedictorian

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`f(x,y,z)=e^(3x)sin(y)cos(z)+sin^(-1)(z^(1/3)/(y^2+z^2))`

Differentiate with respect to y partially

`(delf(x,y,z))/(dely)=f_y=e^(3x)cos(y)cos(z)+1/sqrt(1-(z^(1/3)/(y^2+z^2))^2)del/(dely)(z^(1/3)/(y^2+z^2))`

`=e^(3x)cos(y)cos(z)+(y^2+z^2)/sqrt((y^2+z^2)^2-z^(2/3))(-2yz^(1/3))/(y^2+z^2)^2`

`=e^(3x)cos(y)cos(z)-(2yz^(1/3))/((y^2+z^2)sqrt((y^2+z^2)^2-z^(2/3)))`

Differentiate with respect to z partially

`(delf_y)/(delz)=f_(yz)=-e^(3x)cos(y)sin(z)`

`-2y``{1/3z^(-2/3)(y^2+z^2)sqrt((y^2+z^2)^2-z^(2/3))-``z^(1/3)`

`d/(dz)((y^2+z^2)sqrt((y^2+z^2)^2-z^(2/3)))}``/((y^2+z^2)sqrt((y^2+z^2)^2-z^(2/3)))^2`

`` Differentiate with respect to x partially

`(delf_(yz))/(delx)=-3e^(3x)cos(y)sin(z)` 

`(delf_(yz))/(delx)=f_(yzx)`

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