Question
Function
Find the first partial derivative with respect to x
Find the first partial derivative with respect to y
∂x∂z=2x
Simplify
z=x2+y2
Find the first partial derivative by treating the variable y as a constant and differentiating with respect to x
∂x∂z=∂x∂(x2+y2)
Use differentiation rule ∂x∂(f(x)±g(x))=∂x∂(f(x))±∂x∂(g(x))
∂x∂z=∂x∂(x2)+∂x∂(y2)
Use ∂x∂xn=nxn−1 to find derivative
∂x∂z=2x+∂x∂(y2)
Use ∂x∂(c)=0 to find derivative
∂x∂z=2x+0
Solution
∂x∂z=2x
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Solve the equation
Solve for x
Solve for y
x=z−y2x=−z−y2
Evaluate
z=x2+y2
Swap the sides of the equation
x2+y2=z
Move the expression to the right-hand side and change its sign
x2=z−y2
Take the root of both sides of the equation and remember to use both positive and negative roots
x=±z−y2
Solution
x=z−y2x=−z−y2
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