Question
Solve the equation
Solve for x
Solve for y
x=∣y∣ex=−∣y∣e
Evaluate
x2y2=e
Rewrite the expression
y2x2=e
Divide both sides
y2y2x2=y2e
Divide the numbers
x2=y2e
Take the root of both sides of the equation and remember to use both positive and negative roots
x=±y2e
Simplify the expression
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Evaluate
y2e
To take a root of a fraction,take the root of the numerator and denominator separately
y2e
Simplify the radical expression
∣y∣e
x=±∣y∣e
Solution
x=∣y∣ex=−∣y∣e
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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
x2y2=e
To test if the graph of x2y2=e is symmetry with respect to the origin,substitute -x for x and -y for y
(−x)2(−y)2=e
Evaluate
x2y2=e
Solution
Symmetry with respect to the origin
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Rewrite the equation
r=44ecsc2(2θ)r=−44ecsc2(2θ)
Evaluate
x2y2=e
To convert the equation to polar coordinates,substitute x for rcos(θ) and y for rsin(θ)
(cos(θ)×r)2(sin(θ)×r)2=e
Factor the expression
(cos(θ)sin(θ))2r4=e
Simplify the expression
41sin2(2θ)×r4=e
Divide the terms
r4=sin2(2θ)4e
Simplify the expression
r4=4ecsc2(2θ)
Evaluate the power
r=±44ecsc2(2θ)
Solution
r=44ecsc2(2θ)r=−44ecsc2(2θ)
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Find the first derivative
Find the derivative with respect to x
Find the derivative with respect to y
dxdy=−xy
Calculate
x2y2=e
Take the derivative of both sides
dxd(x2y2)=dxd(e)
Calculate the derivative
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Evaluate
dxd(x2y2)
Use differentiation rules
dxd(x2)×y2+x2×dxd(y2)
Use dxdxn=nxn−1 to find derivative
2xy2+x2×dxd(y2)
Evaluate the derivative
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Evaluate
dxd(y2)
Use differentiation rules
dyd(y2)×dxdy
Use dxdxn=nxn−1 to find derivative
2ydxdy
2xy2+2x2ydxdy
2xy2+2x2ydxdy=dxd(e)
Calculate the derivative
2xy2+2x2ydxdy=0
Move the expression to the right-hand side and change its sign
2x2ydxdy=0−2xy2
Removing 0 doesn't change the value,so remove it from the expression
2x2ydxdy=−2xy2
Divide both sides
2x2y2x2ydxdy=2x2y−2xy2
Divide the numbers
dxdy=2x2y−2xy2
Solution
More Steps

Evaluate
2x2y−2xy2
Cancel out the common factor 2
x2y−xy2
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
xy−y2
Reduce the fraction
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Evaluate
yy2
Use the product rule aman=an−m to simplify the expression
y2−1
Subtract the terms
y1
Simplify
y
x−y
Use b−a=−ba=−ba to rewrite the fraction
−xy
dxdy=−xy
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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=x22y
Calculate
x2y2=e
Take the derivative of both sides
dxd(x2y2)=dxd(e)
Calculate the derivative
More Steps

Evaluate
dxd(x2y2)
Use differentiation rules
dxd(x2)×y2+x2×dxd(y2)
Use dxdxn=nxn−1 to find derivative
2xy2+x2×dxd(y2)
Evaluate the derivative
More Steps

Evaluate
dxd(y2)
Use differentiation rules
dyd(y2)×dxdy
Use dxdxn=nxn−1 to find derivative
2ydxdy
2xy2+2x2ydxdy
2xy2+2x2ydxdy=dxd(e)
Calculate the derivative
2xy2+2x2ydxdy=0
Move the expression to the right-hand side and change its sign
2x2ydxdy=0−2xy2
Removing 0 doesn't change the value,so remove it from the expression
2x2ydxdy=−2xy2
Divide both sides
2x2y2x2ydxdy=2x2y−2xy2
Divide the numbers
dxdy=2x2y−2xy2
Divide the numbers
More Steps

Evaluate
2x2y−2xy2
Cancel out the common factor 2
x2y−xy2
Reduce the fraction
More Steps

Evaluate
x2x
Use the product rule aman=an−m to simplify the expression
x2−11
Subtract the terms
x11
Simplify
x1
xy−y2
Reduce the fraction
More Steps

Evaluate
yy2
Use the product rule aman=an−m to simplify the expression
y2−1
Subtract the terms
y1
Simplify
y
x−y
Use b−a=−ba=−ba to rewrite the fraction
−xy
dxdy=−xy
Take the derivative of both sides
dxd(dxdy)=dxd(−xy)
Calculate the derivative
dx2d2y=dxd(−xy)
Use differentiation rules
dx2d2y=−x2dxd(y)×x−y×dxd(x)
Calculate the derivative
More Steps

Evaluate
dxd(y)
Use differentiation rules
dyd(y)×dxdy
Use dxdxn=nxn−1 to find derivative
dxdy
dx2d2y=−x2dxdy×x−y×dxd(x)
Use dxdxn=nxn−1 to find derivative
dx2d2y=−x2dxdy×x−y×1
Use the commutative property to reorder the terms
dx2d2y=−x2xdxdy−y×1
Any expression multiplied by 1 remains the same
dx2d2y=−x2xdxdy−y
Use equation dxdy=−xy to substitute
dx2d2y=−x2x(−xy)−y
Solution
More Steps

Calculate
−x2x(−xy)−y
Multiply the terms
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Evaluate
x(−xy)
Multiplying or dividing an odd number of negative terms equals a negative
−x×xy
Cancel out the common factor x
−1×y
Multiply the terms
−y
−x2−y−y
Subtract the terms
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Simplify
−y−y
Collect like terms by calculating the sum or difference of their coefficients
(−1−1)y
Subtract the numbers
−2y
−x2−2y
Divide the terms
−(−x22y)
Calculate
x22y
dx2d2y=x22y
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