Z=square root of (x^2+y^2) + xy
Find the partial derivatives:
x, xx, y, yy, xy.
Thanks.
To find the partial derivatives of the given function Z = √(x^2 + y^2) + xy, we need to differentiate Z with respect to each variable while treating the other variables as constants.
1. Partial derivative with respect to x (denoted as ∂Z/∂x):
To differentiate the function with respect to x, we treat y as a constant and differentiate each term separately.
∂Z/∂x = (∂/∂x)(√(x^2 + y^2)) + (∂/∂x)(xy)
The first term requires applying the chain rule. Let's compute it step by step:
Let u = x^2 + y^2
∂Z/∂x = (∂/∂x)(u^(1/2)) + (∂/∂x)(xy)
= (1/2) * (u^(-1/2)) * (∂/∂x)(u) + (∂/∂x)(xy)
= (1/2) * (u^(-1/2)) * (∂/∂x)(x^2 + y^2) + (∂/∂x)(xy)
= (1/2) * (u^(-1/2)) * (2x) + y
= x / √(x^2 + y^2) + y
Hence, ∂Z/∂x = x / √(x^2 + y^2) + y.
2. Partial derivative with respect to y (denoted as ∂Z/∂y):
To differentiate the function with respect to y, we treat x as a constant and differentiate each term separately.
∂Z/∂y = (∂/∂y)(√(x^2 + y^2)) + (∂/∂y)(xy)
We again apply the chain rule to the first term. Let's compute it step by step:
Let u = x^2 + y^2
∂Z/∂y = (∂/∂y)(u^(1/2)) + (∂/∂y)(xy)
= (1/2) * (u^(-1/2)) * (∂/∂y)(u) + (∂/∂y)(xy)
= (1/2) * (u^(-1/2)) * (2y) + x
= y / √(x^2 + y^2) + x
Hence, ∂Z/∂y = y / √(x^2 + y^2) + x.
3. Partial derivative with respect to x, twice (denoted as ∂^2Z/∂x^2):
To compute the second partial derivative with respect to x, we differentiate the result from the first step (∂Z/∂x) with respect to x:
∂^2Z/∂x^2 = (∂/∂x)(∂Z/∂x)
= (∂/∂x)(x / √(x^2 + y^2) + y)
= (√(x^2 + y^2) - 2x^2) / (x^2 + y^2)^(3/2)
Hence, ∂^2Z/∂x^2 = (√(x^2 + y^2) - 2x^2) / (x^2 + y^2)^(3/2).
4. Partial derivative with respect to y, twice (denoted as ∂^2Z/∂y^2):
To compute the second partial derivative with respect to y, we differentiate the result from the second step (∂Z/∂y) with respect to y:
∂^2Z/∂y^2 = (∂/∂y)(∂Z/∂y)
= (∂/∂y)(y / √(x^2 + y^2) + x)
= (√(x^2 + y^2) - 2y^2) / (x^2 + y^2)^(3/2)
Hence, ∂^2Z/∂y^2 = (√(x^2 + y^2) - 2y^2) / (x^2 + y^2)^(3/2).
5. Partial derivative with respect to x and then y (denoted as ∂^2Z/∂x∂y):
To compute the mixed partial derivative, we differentiate the result from the first step (∂Z/∂x) with respect to y:
∂^2Z/∂x∂y = (∂/∂y)(∂Z/∂x)
= (∂/∂y)(x / √(x^2 + y^2) + y)
= (√(x^2 + y^2) - 2xy) / (x^2 + y^2)^(3/2)
Hence, ∂^2Z/∂x∂y = (√(x^2 + y^2) - 2xy) / (x^2 + y^2)^(3/2).
I hope this helps! Let me know if you need any further clarification.