Find all first and second partial derivatives of z with respect to x and y if x^2+4y^2+16z^2−64=0.
x^2+4y^2+16z^2−64 = 0
16z^2 = 64-x^2-4y^2
2z ∂z/∂x = -2x
2z ∂z/dy = -8y
so,
∂z/∂x = -x/z
∂z/∂y = -4y/z
∂^2z/∂x^2 = -(x^2+z^2)/z^3
and you can do the others similarly, using the quotient rule or the product rule
compute the first order partial derivatives of f(x,y) with respect to x and y. assume a constant
To find the first and second partial derivatives of z with respect to x and y, we need to differentiate the given equation with respect to x and y.
Given: x^2 + 4y^2 + 16z^2 − 64 = 0
Step 1: Differentiate the equation with respect to x.
2x + 32z * dz/dx = 0
Step 2: Solve for dz/dx.
dz/dx = -2x / (32z)
Step 3: Differentiate the equation with respect to y.
8y + 32z * dz/dy = 0
Step 4: Solve for dz/dy.
dz/dy = -8y / (32z)
Now, let's find the second partial derivatives.
Step 5: Differentiate dz/dx with respect to x.
d^2z/dx^2 = d/dx (-2x / (32z))
= -2 / (32z)
Step 6: Differentiate dz/dx with respect to y.
d^2z/dydx = d/dy (-2x / (32z))
= 0 (since dz/dx does not depend on y)
Step 7: Differentiate dz/dy with respect to x.
d^2z/dxdy = d/dx (-8y / (32z))
= 0 (since dz/dy does not depend on x)
Step 8: Differentiate dz/dy with respect to y.
d^2z/dy^2 = d/dy (-8y / (32z))
= -8 / (32z)
= -1 / (4z)
Therefore, the first partial derivatives are:
dz/dx = -2x / (32z)
dz/dy = -8y / (32z)
The second partial derivatives are:
d^2z/dx^2 = -2 / (32z)
d^2z/dydx = 0
d^2z/dxdy = 0
d^2z/dy^2 = -1 / (4z)
To find the first and second partial derivatives of z with respect to x and y, we start by differentiating the equation x^2 + 4y^2 + 16z^2 - 64 = 0 with respect to x and y.
Differentiating with respect to x:
2x + 32z * dz/dx = 0
Differentiating with respect to y:
8y + 32z * dz/dy = 0
From these two equations, we need to find dz/dx and dz/dy.
To find dz/dx, we can rearrange the equation 2x + 32z * dz/dx = 0 to solve for dz/dx:
dz/dx = -2x / (32z)
To find dz/dy, we can rearrange the equation 8y + 32z * dz/dy = 0 to solve for dz/dy:
dz/dy = -8y / (32z)
Now, let's find the second partial derivatives.
To find d^2z/dx^2, we differentiate dz/dx with respect to x:
d^2z/dx^2 = -2 / (32z)
To find d^2z/dy^2, we differentiate dz/dy with respect to y:
d^2z/dy^2 = -8 / (32z)
Finally, to find d^2z/dxdy, we differentiate dz/dx with respect to y:
d^2z/dxdy = (-2 / (32z)) * dz/dy
= (-2 / (32z)) * (-8y / (32z))
= 16y / (256z^2)
Therefore, the first and second partial derivatives of z with respect to x and y are:
dz/dx = -2x / (32z)
dz/dy = -8y / (32z)
d^2z/dx^2 = -2 / (32z)
d^2z/dy^2 = -8 / (32z)
d^2z/dxdy = 16y / (256z^2)