A Anum New member Joined May 5, 2012 Messages 2 May 5, 2012 #1 [h=1]Hi!!! If z= f(x,xy)+g(xy), what will be the value of Zxy (The partial derivative of z with respect to x and y)? Looking forward for a kind response Thanks in advance[/h]
[h=1]Hi!!! If z= f(x,xy)+g(xy), what will be the value of Zxy (The partial derivative of z with respect to x and y)? Looking forward for a kind response Thanks in advance[/h]
H HallsofIvy Elite Member Joined Jan 27, 2012 Messages 7,760 May 6, 2012 #2 Anum said: Hi!!! If z= f(x,xy)+g(xy), what will be the value of Zxy (The partial derivative of z with respect to x and y)? Looking forward for a kind response Thanks in advance Click to expand... Let u= xy so z= f(x,u)+ g(x,u). Then \(\displaystyle z_x= f_x+ f_uu_x+ g_x+ g_uu_x= f_x+ yf_u+ g_x+ yg_u\) Differentiating with respect to y now, \(\displaystyle z_{xy}= f_{xu}u_y+ f_u+ yf_{uu}u_y+ g_{xu}u_y+ g_u+ yg_{uu}u_y\)\(\displaystyle = xf_{xu}+ f_u+ xyf_{uu}+ xg_{xu}+ g_u+ xyg_{uu}\)
Anum said: Hi!!! If z= f(x,xy)+g(xy), what will be the value of Zxy (The partial derivative of z with respect to x and y)? Looking forward for a kind response Thanks in advance Click to expand... Let u= xy so z= f(x,u)+ g(x,u). Then \(\displaystyle z_x= f_x+ f_uu_x+ g_x+ g_uu_x= f_x+ yf_u+ g_x+ yg_u\) Differentiating with respect to y now, \(\displaystyle z_{xy}= f_{xu}u_y+ f_u+ yf_{uu}u_y+ g_{xu}u_y+ g_u+ yg_{uu}u_y\)\(\displaystyle = xf_{xu}+ f_u+ xyf_{uu}+ xg_{xu}+ g_u+ xyg_{uu}\)
