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\fdz Find each partial derivative needed to use the Chain Rule to find dz ddx - sin v = -6e* Iny Ax COS V dz -6e X dy E U COS V ay y av dz 6u cos 2 V Using either method, = 6u In (usin v) sin v- sin v (Type an expression using u and v as the variables.) dz 71/3 (b) = du - 3 1+ In 2 (Type an exact answer.) 7 , dz E (Type an exact answer.) 7.Let w = f(u) + g(v), where u =x + iy, v=x - iy, and i = 1 -1. Show that w satisfies the Laplace equation Wxx + Wyy =0 if all the necessary functions are differentiable. ow Recall that wx is the same as - ax2 . First find wx which is the same as -ax WX ax )(1) + (wv) (1) (Simplify your answer.) Next, find Wxx owx owx WXx = du a((Wu) + ( Wv ) ) a (( Wu ) + ( Wv ) ) (1) du Note that Wyy =0 since wy will be a function in v only and so Wy will be 0. Similarly, Wuv will be zero. Using this information simplify Wxx. wxx aw aw Now, find wyy. Recall that Wyy is the same as -2 . First find wy which is the same as dy. dwy du dwy W yy du ay ay d((iwu ) - (iwv) ) ac( wu) - (:wo) ( -2) du ov Recall that Wvu = Wuv = 0. Using this information simplify wyy. Finally, add the results. Wxx + Wyy Therefore, does w satisfy the Laplace equation? O No O YesThe temperature T = T(x,y) in C at point (x,y) satisfies T, (1,4) = 4 and Ty(1,4) = -6. If x=e-# cm and y = 4 + Int cm, find the rate at which the temperature T changes when t = 1 sec. When t= 1 sec, the temperature changes at a rate of "C per sec