Question
Function
Find the first partial derivative with respect to x
Find the first partial derivative with respect to y
∂x∂z=4xy2
Simplify
z=2x2y2
Find the first partial derivative by treating the variable y as a constant and differentiating with respect to x
∂x∂z=∂x∂(2x2y2)
Use differentiation rule ∂x∂(cf(x))=c×∂x∂(f(x))
∂x∂z=2y2×∂x∂(x2)
Use ∂x∂xn=nxn−1 to find derivative
∂x∂z=2y2×2x
Solution
∂x∂z=4xy2
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Solve the equation
Solve for x
Solve for y
x=−2∣y∣2z,y=0x=2∣y∣2z,y=0
Evaluate
z=2x2y2
Rewrite the expression
z=2y2x2
Swap the sides of the equation
2y2x2=z
Divide both sides
2y22y2x2=2y2z
Divide the numbers
x2=2y2z
Take the root of both sides of the equation and remember to use both positive and negative roots
x=±2y2z
Simplify the expression
More Steps

Evaluate
2y2z
To take a root of a fraction,take the root of the numerator and denominator separately
2y2z
Simplify the radical expression
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Evaluate
2y2
Rewrite the expression
2×y2
Simplify the root
2×∣y∣
2×∣y∣z
Multiply by the Conjugate
2×∣y∣×2z×2
Calculate
2∣y∣z×2
Calculate
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Evaluate
z×2
The product of roots with the same index is equal to the root of the product
z×2
Calculate the product
2z
2∣y∣2z
x=±2∣y∣2z
Separate the equation into 2 possible cases
x=2∣y∣2zx=−2∣y∣2z
Calculate
{x=−2∣y∣2zy=0{x=2∣y∣2zy=0
Solution
x=−2∣y∣2z,y=0x=2∣y∣2z,y=0
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