Find (d^2*z)/(dx^2)+(d^2*z)/(dy^2).

Question

z=(e^x)(cos(y)). Find (d^2*z)/(dx^2)+(d^2*z)/(dy^2).

Answer 1

∂z/∂x = (e^x)(cos(y))

∂²z/∂x² = (e^x)(cos(y))

∂z/∂y = (e^x)(-sin(y))

∂²z/∂y² = (e^x)(-cos(y))

So, ∂²z/∂x² + ∂²z/∂y² = (e^x)(cos(y)) + (e^x)(-cos(y)) = 0 <==Answer

Answer 2

When you take a partial derivative with respect to a variable, in this case x, the other variables are treated as constants and do not change.

∂z/∂x = ∂/∂x(e^x*cos(y))=>e^x*cos(y)

the derivative of e^x is itself e^x times the derivative of x which is one. e^x*1=e^x and cos(y) tags along for the ride, because it's a partial with respect to x, not y. 

so the second derivative with respect to x will have the same answer.

∂^2z/∂x^2 = e^x*cos(y).

So now, to find the first partial with respect to y, e^x becomes a constant and the derivative of cos(y) is just -sin(y). 

∂z/∂y = e^x*(-sin(y))

for the second partial with respect to y, we treat -e^x as a constant and derive sin(y), which is cos(y)

∂^2z/∂y^2 = -e^x*cos(y)

Now adding together we get ∂^2z/∂x^2 + ∂^2z/∂y^2 = e^x*cos(y) + (-e^x*cos(y)) = 0