When you compute #(del(z(x,y)))/(delx)#, you treat #y# and any function of #y# as if it were a constant; this allows you to move it outside the derivative as you would any constant, then differentiate the remaining function as you would a full derivative.
#(del(z(x,y)))/(delx) = (del(e^(-3x)cos(y)))/(delx)#
Treat #cos(y)# as if it were a constant:
#(del(z(x,y)))/(delx) = cos(y)(del(e^(-3x)))/(delx)#
Then differentiate the remaining function as if it were a full derivative:
#(del(z(x,y)))/(delx) = cos(y)(d(e^(-3x)))/(dx)#
#(del(z(x,y)))/(delx) = cos(y)(-3e^(-3x))#
We repeat the process for the second derivative:
#(del^2(z(x,y)))/(delx^2) = (del(cos(y)(-3e^(-3x))))/(delx)#
Move the factors #cos(y)# and #-3# outside the derivative:
#(del^2(z(x,y)))/(delx^2) = -3cos(y)(del(e^(-3x)))/(delx)#
Differentiate the remaining functions as if it were a full derivative:
#(del^2(z(x,y)))/(delx^2) = -3cos(y)(d(e^(-3x)))/(dx)#
#(del^2(z(x,y)))/(delx^2) = -3cos(y)(-3e^(-3x))#
Simplify:
#(del^2(z(x,y)))/(delx^2) = 9cos(y)e^(-3x)#
