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[Anum]

[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]

 

[HallsofIvy]

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

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, \(\displaystyle z_{xy}= f_{xu}u_y+ f_u+ yf_{uu}u_y+ g_{xu}u_y+ g_u+ yg_{uu}u_y\)\(\displaystyle = xf_{xu}+ f_u+ xyf_{uu}+ xg_{xu}+ g_u+ xyg_{uu}\)