In Exercises 3–10, use the Chain Rule to calculate the partial derivatives. Express the answer in terms of the independent varia
bles. 3. ∂f ∂s , ∂f ∂r ; f (x, y, z) = xy + z2, x = s2, y = 2rs, z = r2 4. ∂f ∂r , ∂f ∂t ; f (x, y, z) = xy + z2, x = r + s − 2t, y = 3rt, z = s2
1 answer:
I'll use subscript notation for brevity, i.e.
![\dfrac{\partial f}{\partial x}=f_x](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20x%7D%3Df_x)
3.
![f(x,y,z)=xy+z^2\implies\begin{cases}f_x=y\\f_y=x\\f_z=2z\end{cases}](https://tex.z-dn.net/?f=f%28x%2Cy%2Cz%29%3Dxy%2Bz%5E2%5Cimplies%5Cbegin%7Bcases%7Df_x%3Dy%5C%5Cf_y%3Dx%5C%5Cf_z%3D2z%5Cend%7Bcases%7D)
![x(r,s)=s^2\implies\begin{cases}x_r=0\\x_s=2s\end{cases}](https://tex.z-dn.net/?f=x%28r%2Cs%29%3Ds%5E2%5Cimplies%5Cbegin%7Bcases%7Dx_r%3D0%5C%5Cx_s%3D2s%5Cend%7Bcases%7D)
![y(r,s)=2rs\implies\begin{cases}y_r=2s\\y_s=2r\end{cases}](https://tex.z-dn.net/?f=y%28r%2Cs%29%3D2rs%5Cimplies%5Cbegin%7Bcases%7Dy_r%3D2s%5C%5Cy_s%3D2r%5Cend%7Bcases%7D)
![z(r,s)=r^2\implies\begin{cases}z_r=2r\\z_s=0\end{cases}](https://tex.z-dn.net/?f=z%28r%2Cs%29%3Dr%5E2%5Cimplies%5Cbegin%7Bcases%7Dz_r%3D2r%5C%5Cz_s%3D0%5Cend%7Bcases%7D)
By the chain rule,
![f_r=f_xx_r+f_yy_r+f_zz_r=2xs+4zr=2s^3+4r^3](https://tex.z-dn.net/?f=f_r%3Df_xx_r%2Bf_yy_r%2Bf_zz_r%3D2xs%2B4zr%3D2s%5E3%2B4r%5E3)
![f_s=f_xx_s+f_yy_s+f_zz_s=2ys+2xr=6rs^2](https://tex.z-dn.net/?f=f_s%3Df_xx_s%2Bf_yy_s%2Bf_zz_s%3D2ys%2B2xr%3D6rs%5E2)
4.
![x(r,s,t)=r+s-2t\implies\begin{cases}x_r=1\\x_s=1\\x_t=-2\end{cases}](https://tex.z-dn.net/?f=x%28r%2Cs%2Ct%29%3Dr%2Bs-2t%5Cimplies%5Cbegin%7Bcases%7Dx_r%3D1%5C%5Cx_s%3D1%5C%5Cx_t%3D-2%5Cend%7Bcases%7D)
![y(r,s,t)=3rt\implies\begin{cases}y_r=3t\\y_s=0\\y_t=3r\end{cases}](https://tex.z-dn.net/?f=y%28r%2Cs%2Ct%29%3D3rt%5Cimplies%5Cbegin%7Bcases%7Dy_r%3D3t%5C%5Cy_s%3D0%5C%5Cy_t%3D3r%5Cend%7Bcases%7D)
![z(r,s,t)=s^2\implies\begin{cases}z_r=0\\z_s=2s\\z_t=0\end{cases}](https://tex.z-dn.net/?f=z%28r%2Cs%2Ct%29%3Ds%5E2%5Cimplies%5Cbegin%7Bcases%7Dz_r%3D0%5C%5Cz_s%3D2s%5C%5Cz_t%3D0%5Cend%7Bcases%7D)
By the chain rule,
![f_r=f_xx_r+f_yy_r+f_zz_r=y+3xt=3rt+3r+3s-6t^2](https://tex.z-dn.net/?f=f_r%3Df_xx_r%2Bf_yy_r%2Bf_zz_r%3Dy%2B3xt%3D3rt%2B3r%2B3s-6t%5E2)
![f_s=f_xx_s+f_yy_s+f_zz_s=y+4zs=3rt+4s^3](https://tex.z-dn.net/?f=f_s%3Df_xx_s%2Bf_yy_s%2Bf_zz_s%3Dy%2B4zs%3D3rt%2B4s%5E3)
![f_t=f_xx_t+f_yy_t+f_zz_t=-2y+3xr=3r+3s-12rt](https://tex.z-dn.net/?f=f_t%3Df_xx_t%2Bf_yy_t%2Bf_zz_t%3D-2y%2B3xr%3D3r%2B3s-12rt)
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