Question
Solve the equation
Solve for x
Solve for y
Solve for z
x=2y316y2+4zy2
Evaluate
2x3y−z=4
Rewrite the expression
2yx3−z=4
Move the expression to the right-hand side and change its sign
2yx3=4+z
Divide both sides
2y2yx3=2y4+z
Divide the numbers
x3=2y4+z
Take the 3-th root on both sides of the equation
3x3=32y4+z
Calculate
x=32y4+z
Simplify the root
More Steps

Evaluate
32y4+z
To take a root of a fraction,take the root of the numerator and denominator separately
32y34+z
Multiply by the Conjugate
32y×322y234+z×322y2
Calculate
2y34+z×322y2
Calculate
More Steps

Evaluate
34+z×322y2
The product of roots with the same index is equal to the root of the product
3(4+z)×22y2
Calculate the product
324y2+22zy2
2y324y2+22zy2
Calculate
2y316y2+22zy2
x=2y316y2+22zy2
Solution
x=2y316y2+4zy2
Show Solution

Find the partial derivative
Find ∂x∂z by differentiating the equation directly
Find ∂y∂z by differentiating the equation directly
∂x∂z=6x2y
Evaluate
2x3y−z=4
Find ∂x∂z by taking the derivative of both sides with respect to x
∂x∂(2x3y−z)=∂x∂(4)
Use differentiation rule ∂x∂(f(x)±g(x))=∂x∂(f(x))±∂x∂(g(x))
∂x∂(2x3y)−∂x∂(z)=∂x∂(4)
Evaluate
More Steps

Evaluate
∂x∂(2x3y)
Use differentiation rule ∂x∂(cf(x))=c×∂x∂(f(x))
2y×∂x∂(x3)
Use ∂x∂xn=nxn−1 to find derivative
2y×3x2
Multiply the terms
6x2y
6x2y−∂x∂(z)=∂x∂(4)
Evaluate
6x2y−∂x∂z=∂x∂(4)
Find the partial derivative
6x2y−∂x∂z=0
Move the expression to the right-hand side and change its sign
−∂x∂z=0−6x2y
Removing 0 doesn't change the value,so remove it from the expression
−∂x∂z=−6x2y
Solution
∂x∂z=6x2y
Show Solution
