give that z = f(x,y). use partial derivatives to find δz/δx and δz/δy for
xe^xz + z(xy+1)^3-sin x cos^2y=xy/z
present your final answer in the simplest form..
please solve this problem....tq
Sure, let's solve the problem using partial derivatives.
To find δz/δx, we need to differentiate the equation with respect to x while treating y as a constant. The given equation is:
xe^xz + z(xy + 1)^3 - sin(x)cos^2(y) = xy/z
Differentiating both sides with respect to x:
e^xz + xze^xz + 3z(xy + 1)^2 * y - cos(x)cos^2(y) = y/z
Now, let's simplify this equation and isolate δz/δx:
e^xz + xze^xz - cos(x)cos^2(y) = y/z - 3z(xy + 1)^2 * y
δz/δx = y/z - 3z(xy + 1)^2 * y - xze^xz + cos(x)cos^2(y)
Similarly, to find δz/δy, we differentiate the equation with respect to y while treating x as a constant:
e^xz + 3z(xy + 1)^2 * x - 2sin(x)cos(y)cos(y) = x/z
Simplifying and isolating δz/δy:
δz/δy = x/z - 3z(xy + 1)^2 * x + 2sin(x)cos(y)cos(y)
So, the partial derivatives are:
δz/δx = y/z - 3z(xy + 1)^2 * y - xze^xz + cos(x)cos^2(y)
δz/δy = x/z - 3z(xy + 1)^2 * x + 2sin(x)cos(y)cos(y)
I hope this clarifies the solution for you!
To find the partial derivatives of z with respect to x and y, we will use the given function:
z = xe^xz + z(xy+1)^3 - sin(x) cos^2(y) - xy/z
Step 1: Find ∂z/∂x by differentiating the function with respect to x while treating y as a constant.
∂z/∂x = e^xz + xz * e^xz + 3z * (xy + 1)^2
Step 2: Simplify the expression.
∂z/∂x = e^xz + xz * e^xz + 3z * (xy + 1)^2
Step 3: Find ∂z/∂y by differentiating the function with respect to y while treating x as a constant.
∂z/∂y = -3z * sin(x) * 2 * cos(y) * (-sin(y))
Step 4: Simplify the expression.
∂z/∂y = 6z * sin(x) * cos(y) * sin(y)
Hence, the partial derivatives are:
∂z/∂x = e^xz + xz * e^xz + 3z * (xy + 1)^2
∂z/∂y = 6z * sin(x) * cos(y) * sin(y)
To find the partial derivatives of z with respect to x and y, we can differentiate the given equation with respect to x and y while treating y as a constant when differentiating with respect to x, and vice versa.
Given: z = f(x, y) = xe^xz + z(xy+1)^3 - sin x cos^2y - xy/z
Step 1: Calculate δz/δx:
To find δz/δx, we differentiate the equation with respect to x while treating y as a constant.
Differentiating the equation with respect to x:
d/dx (xe^xz + z(xy+1)^3 - sin(x)cos^2(y)) = d/dx (xye^xz + z(xy+1)^3) - cos(y)cos^2(y)
Now, let's find the partial derivative of each term:
1. d/dx (xye^xz) = y(e^xz + xe^xz)
2. d/dx (z(xy+1)^3) = z * 3(xy+1)^2 * y
3. d/dx (-sin(x)cos^2(y)) = -cos(x)cos^2(y)
Therefore, δz/δx = y(e^xz + xe^xz) + 3z(xy+1)^2y - cos(x)cos^2(y)
Step 2: Calculate δz/δy
To find δz/δy, we differentiate the equation with respect to y while treating x as a constant.
Differentiating the equation with respect to y:
d/dy (xe^xz + z(xy+1)^3 - sin(x)cos^2(y)) = d/dy (z(xy+1)^3) - sin(x)2cos(y)(-sin(y))
Now, let's find the partial derivative of each term:
1. d/dy (z(xy+1)^3) = z * 3(xy+1)^2 * x
2. d/dy (-sin(x)cos^2(y)) = -2sin(x)cos^3(y)
Therefore, δz/δy = 3z(xy+1)^2x - 2sin(x)cos^3(y)
So, the final answers for the partial derivatives are:
δz/δx = y(e^xz + xe^xz) + 3z(xy+1)^2y - cos(x)cos^2(y)
δz/δy = 3z(xy+1)^2x - 2sin(x)cos^3(y)
Please note that these expressions can be further simplified if desired.