Question
Solve the equation
Solve for x
Solve for y
x=0
Evaluate
2x2y=0
Rewrite the expression
2yx2=0
Rewrite the expression
x2=0
Solution
x=0
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Testing for symmetry
Testing for symmetry about the origin
Testing for symmetry about the x-axis
Testing for symmetry about the y-axis
Symmetry with respect to the origin
Evaluate
2x2y=0
To test if the graph of 2x2y=0 is symmetry with respect to the origin,substitute -x for x and -y for y
2(−x)2(−y)=0
Evaluate
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Evaluate
2(−x)2(−y)
Any expression multiplied by 1 remains the same
−2(−x)2y
Multiply the terms
−2x2y
−2x2y=0
Solution
Symmetry with respect to the origin
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Rewrite the equation
r=0θ=2kπ,k∈Z
Evaluate
2x2y=0
To convert the equation to polar coordinates,substitute x for rcos(θ) and y for rsin(θ)
2(cos(θ)×r)2sin(θ)×r=0
Factor the expression
2cos2(θ)sin(θ)×r3=0
Separate into possible cases
r3=02cos2(θ)sin(θ)=0
Evaluate
r=02cos2(θ)sin(θ)=0
Solution
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Evaluate
2cos2(θ)sin(θ)=0
Elimination the left coefficient
cos2(θ)sin(θ)=0
Separate the equation into 2 possible cases
cos2(θ)=0sin(θ)=0
Solve the equation
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Evaluate
cos2(θ)=0
The only way a power can be 0 is when the base equals 0
cos(θ)=0
Use the inverse trigonometric function
θ=arccos(0)
Calculate
θ=2π
Add the period of kπ,k∈Z to find all solutions
θ=2π+kπ,k∈Z
θ=2π+kπ,k∈Zsin(θ)=0
Solve the equation
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Evaluate
sin(θ)=0
Use the inverse trigonometric function
θ=arcsin(0)
Calculate
θ=0
Add the period of kπ,k∈Z to find all solutions
θ=kπ,k∈Z
θ=2π+kπ,k∈Zθ=kπ,k∈Z
Find the union
θ=2kπ,k∈Z
r=0θ=2kπ,k∈Z
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Find the first derivative
Find the derivative with respect to x
Find the derivative with respect to y
dxdy=−x2y
Calculate
2x2y=0
Take the derivative of both sides
dxd(2x2y)=dxd(0)
Calculate the derivative
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Evaluate
dxd(2x2y)
Use differentiation rules
dxd(2x2)×y+2x2×dxd(y)
Evaluate the derivative
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Evaluate
dxd(2x2)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
2×dxd(x2)
Use dxdxn=nxn−1 to find derivative
2×2x
Multiply the terms
4x
4xy+2x2×dxd(y)
Evaluate the derivative
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Evaluate
dxd(y)
Use differentiation rules
dyd(y)×dxdy
Use dxdxn=nxn−1 to find derivative
dxdy
4xy+2x2dxdy
4xy+2x2dxdy=dxd(0)
Calculate the derivative
4xy+2x2dxdy=0
Move the expression to the right-hand side and change its sign
2x2dxdy=0−4xy
Removing 0 doesn't change the value,so remove it from the expression
2x2dxdy=−4xy
Divide both sides
2x22x2dxdy=2x2−4xy
Divide the numbers
dxdy=2x2−4xy
Solution
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Evaluate
2x2−4xy
Cancel out the common factor 2
x2−2xy
Reduce the fraction
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Evaluate
x2x
Use the product rule aman=an−m to simplify the expression
x2−11
Subtract the terms
x11
Simplify
x1
x−2y
Use b−a=−ba=−ba to rewrite the fraction
−x2y
dxdy=−x2y
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Find the second derivative
Find the second derivative with respect to x
Find the second derivative with respect to y
dx2d2y=x26y
Calculate
2x2y=0
Take the derivative of both sides
dxd(2x2y)=dxd(0)
Calculate the derivative
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Evaluate
dxd(2x2y)
Use differentiation rules
dxd(2x2)×y+2x2×dxd(y)
Evaluate the derivative
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Evaluate
dxd(2x2)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
2×dxd(x2)
Use dxdxn=nxn−1 to find derivative
2×2x
Multiply the terms
4x
4xy+2x2×dxd(y)
Evaluate the derivative
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Evaluate
dxd(y)
Use differentiation rules
dyd(y)×dxdy
Use dxdxn=nxn−1 to find derivative
dxdy
4xy+2x2dxdy
4xy+2x2dxdy=dxd(0)
Calculate the derivative
4xy+2x2dxdy=0
Move the expression to the right-hand side and change its sign
2x2dxdy=0−4xy
Removing 0 doesn't change the value,so remove it from the expression
2x2dxdy=−4xy
Divide both sides
2x22x2dxdy=2x2−4xy
Divide the numbers
dxdy=2x2−4xy
Divide the numbers
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Evaluate
2x2−4xy
Cancel out the common factor 2
x2−2xy
Reduce the fraction
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Evaluate
x2x
Use the product rule aman=an−m to simplify the expression
x2−11
Subtract the terms
x11
Simplify
x1
x−2y
Use b−a=−ba=−ba to rewrite the fraction
−x2y
dxdy=−x2y
Take the derivative of both sides
dxd(dxdy)=dxd(−x2y)
Calculate the derivative
dx2d2y=dxd(−x2y)
Use differentiation rules
dx2d2y=−x2dxd(2y)×x−2y×dxd(x)
Calculate the derivative
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Evaluate
dxd(2y)
Simplify
2×dxd(y)
Calculate
2dxdy
dx2d2y=−x22dxdy×x−2y×dxd(x)
Use dxdxn=nxn−1 to find derivative
dx2d2y=−x22dxdy×x−2y×1
Use the commutative property to reorder the terms
dx2d2y=−x22xdxdy−2y×1
Any expression multiplied by 1 remains the same
dx2d2y=−x22xdxdy−2y
Use equation dxdy=−x2y to substitute
dx2d2y=−x22x(−x2y)−2y
Solution
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Calculate
−x22x(−x2y)−2y
Multiply
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Multiply the terms
2x(−x2y)
Any expression multiplied by 1 remains the same
−2x×x2y
Multiply the terms
−4y
−x2−4y−2y
Subtract the terms
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Simplify
−4y−2y
Collect like terms by calculating the sum or difference of their coefficients
(−4−2)y
Subtract the numbers
−6y
−x2−6y
Divide the terms
−(−x26y)
Calculate
x26y
dx2d2y=x26y
Show Solution
