Suppose xy^3z^2+133=2xy−z . Compute ∂z/∂x and ∂z/∂y at the point (2,−2,3).
To compute ∂z/∂x and ∂z/∂y at the point (2,-2,3), we need to find the partial derivatives of z with respect to x and y.
Given the equation: xy^3z^2 + 133 = 2xy - z
Step 1: Differentiate both sides of the equation with respect to x while treating y and z as constants.
Partial derivative of xy^3z^2 with respect to x:
Differentiating xy^3z^2 with respect to x using the product rule, we get:
(1 * y^3z^2) + (x * 3y^2z^2 * ∂z/∂x) = 2y - 0 (derivative of z with respect to x is 0 as z is not dependent on x)
Simplifying the equation, we get:
y^3z^2 + 3xy^2z^2 * ∂z/∂x = 2y
Step 2: Differentiate both sides of the equation with respect to y while treating x and z as constants.
Partial derivative of xy^3z^2 with respect to y:
Differentiating xy^3z^2 with respect to y using the product rule, we get:
(x * 3y^2z^2 * ∂z/∂y) + (xy^3 * 2z * ∂z/∂y) = 0
Simplifying the equation, we get:
3xyz^2 * ∂z/∂y + 2xyz^2 * ∂z/∂y = 0
Step 3: Substitute the values of x, y, and z from the given point (2, -2, 3) into the above equations.
Using the point (2, -2, 3), we have:
y = -2, x = 2, z = 3
Substituting the values from the point into the equations, we get:
-2^3 * 3^2 + 3*2*(-2)^2 * ∂z/∂x = 2*(-2)
-8 * 9 + 3*2*4 * ∂z/∂x = -4
-72 + 24 * ∂z/∂x = -4
24 * ∂z/∂x = 72 - 4
24 * ∂z/∂x = 68
∂z/∂x = 68/24
∂z/∂x = 17/6
Similarly, substituting the values from the point into the second equation, we get:
3*2*3^2 * ∂z/∂y + 2*2*3^2 * ∂z/∂y = 0
54 * ∂z/∂y + 36 * ∂z/∂y = 0
90 * ∂z/∂y = 0
∂z/∂y = 0
Therefore, at the point (2, -2, 3), ∂z/∂x = 17/6 and ∂z/∂y = 0.