f(x,y)=2xe^y-3ye^-x derive second order
I think you mean find the second derivative. Do you mean partial derivatives? There are three.
Fx = 2e^y + 3ye^-x
Fy = 2xe^y - 3e^-x
Fxx = -3ye^-x
Fyy = 2xe^y
Fxy = Fyx = 2e^y + 3e^-x
To find the second-order derivative of the given function f(x, y) = 2xe^y - 3ye^(-x), we need to differentiate it twice with respect to both x and y separately. Here is the step-by-step process:
Step 1: Differentiate with respect to x.
To find the first derivative with respect to x, treat y as a constant and differentiate each term of the function:
∂f/∂x = ∂/∂x (2xe^y) - ∂/∂x (3ye^(-x))
Differentiating each term:
∂/∂x (2xe^y) = 2e^y + 2xe^y * ∂/∂x (y)
= 2e^y + 2xye^y
∂/∂x (3ye^(-x)) = -3e^(-x) * ∂/∂x (y) - 3y * ∂/∂x (e^(-x))
= -3e^(-x)* ∂/∂x (y) - 3ye^(-x) * ∂/∂x (-x)
= -3e^(-x) * ∂/∂x (y) + 3ye^(-x)
So ∂f/∂x = 2e^y + 2xye^y - 3e^(-x) * ∂/∂x (y) + 3ye^(-x)
Step 2: Differentiate ∂f/∂x with respect to x.
To find the second derivative of ∂f/∂x with respect to x, differentiate each term again:
∂²f/∂x² = ∂/∂x ( 2e^y + 2xye^y - 3e^(-x) * ∂/∂x (y) + 3ye^(-x) )
Differentiating each term:
∂/∂x ( 2e^y + 2xye^y ) = 2e^y * ∂/∂x (y) + 2ye^y + 2xye^y * ∂/∂x (y) + 2ye^y
= 2e^y * ∂/∂x (y) + 2xye^y + 2xye^y * ∂/∂x (y) + 2ye^y
= 2e^y * ∂/∂x (y) + 2xye^y * ∂/∂x (y) + 2xye^y + 2ye^y
∂/∂x ( -3e^(-x) * ∂/∂x (y) + 3ye^(-x) ) = 3e^(-x) * ∂²/∂x² (y) + 3ye^(-x) * ∂/∂x (-x) - 3e^(-x) * (∂/∂x (y))² - 3e^(-x)
= 3e^(-x) * ∂²/∂x² (y) - 3ye^(-x) - 3e^(-x) * (∂/∂x (y))² - 3e^(-x)
Therefore, ∂²f/∂x² = 2e^y * ∂/∂x (y) + 2xye^y * ∂/∂x (y) + 2xye^y + 2ye^y + 3e^(-x) * ∂²/∂x² (y) - 3ye^(-x) - 3e^(-x) * (∂/∂x (y))² - 3e^(-x)
Keep in mind that ∂²f/∂x² represents the second derivative of f(x, y) with respect to x.
Please note that we have not taken the second derivative with respect to y yet. If you want the second derivative with respect to y as well, please let me know and I can assist you further.
To find the second-order derivative of the function f(x, y) = 2xe^y - 3ye^-x, we need to take the derivative twice. Let's start by finding the first-order derivative with respect to x.
Step 1: Find the first-order derivative with respect to x.
To do this, we treat y as a constant and differentiate each term separately with respect to x.
The derivative of 2xe^y with respect to x is 2e^y.
The derivative of -3ye^-x with respect to x is 3ye^-x.
So the first-order derivative of f(x, y) with respect to x is:
f'x(x, y) = 2e^y + 3ye^-x.
Step 2: Find the second-order derivative with respect to x.
To find the second-order derivative, we need to differentiate the first-order derivative obtained in Step 1 with respect to x.
Differentiating 2e^y with respect to x gives us 0 since e^y does not depend on x.
Differentiating 3ye^-x with respect to x gives us -3ye^-x.
Therefore, the second-order derivative of f(x, y) with respect to x is:
f''xx(x, y) = 0 - 3ye^-x = -3ye^-x.
And that's the second-order derivative of the function f(x, y) = 2xe^y - 3ye^-x with respect to x.