w = (e^2x)(siny)?

Question

w = (e^2x)(siny)

how do i find d^2w / dx^2 + d^2w / dy^2??


my answer doesnt seem to make much sense...

Answer 1

What you want is partial derivatives. With partial derivatives, you differentiate each variable independently, holding all other variables as constant.

dw/dx = 2*e^(2x) * sin(y)
You treat the y as a constant, so there is no y' or dy/dx.
y is not a function of x. x and y are both independent variables and w is a function of x and y.

d2w/dx2 = 4*e^(2x) * sin(y)

dw/dy = e^(2x) * cosy
d2w/dy2 = -e^(2x) * sin(y)

Answer 2

I assume that your derivatives are in fact partial derivatives (with the round "d").

For a partial derivative, you simply calculate the derivative the usual way, as if every other variable is constant (they are constant, for the purpose of calculating the partial derivative).

For example, the first partial derivative with respect to x is the same as if sin(y) is constant and you derivate only the part dependent on x (the exponential):
dw/dx = 2 e^2x sin(y)
Derivate once again to find the second partial derivative with respect to x, and use the same principle to find the second derivative with respect to y, then sum them up...

Just as a note, for a function of 2 variables, the sum d^2w / dx^2 + d^2w / dy^2 is called Laplacean

Answer 3

Experience dictates that you probably meant to write W=e^(2x) sin y and not W=e² x sin y (which is what you actually wrote -- remember that exponentiation is first in the order of operations). If this assumption is incorrect, then I apologize. Anyway, you have a very nice function for partial derivatives: it is the product of two functions in one variable. This means that, for instance, when differentiating W with respect to x, sin y is a constant coefficient (w.r.t. x) and thus you do not require the product rule. Similarly with e^(2x) when differentiating w.r.t. y.

On to the derivation:

∂w/∂x = 2e^(2x) sin y
∂²w/∂x² = 4e^(2x) sin y

∂w/∂y = e^(2x) cos y
∂²w/∂y² = -e^(2x) sin y

∂²w/∂x² + ∂²w/∂y² = 4 e^(2x) sin y - e^(2x) sin y = 3 e^(2x) sin y

And we are done.