Question
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
x4−y4=xy
To test if the graph of x4−y4=xy is symmetry with respect to the origin,substitute -x for x and -y for y
(−x)4−(−y)4=−x(−y)
Evaluate
More Steps

Evaluate
(−x)4−(−y)4
Rewrite the expression
x4−(−y)4
Rewrite the expression
x4−y4
x4−y4=−x(−y)
Multiplying or dividing an even number of negative terms equals a positive
x4−y4=xy
Solution
Symmetry with respect to the origin
Show Solution

Rewrite the equation
r=0r=22tan(2θ)r=−22tan(2θ)
Evaluate
x4−y4=xy
Move the expression to the left side
x4−y4−xy=0
To convert the equation to polar coordinates,substitute x for rcos(θ) and y for rsin(θ)
(cos(θ)×r)4−(sin(θ)×r)4−cos(θ)×rsin(θ)×r=0
Factor the expression
(cos4(θ)−sin4(θ))r4−cos(θ)sin(θ)×r2=0
Simplify the expression
(cos4(θ)−sin4(θ))r4−21sin(2θ)×r2=0
Factor the expression
r2((cos4(θ)−sin4(θ))r2−21sin(2θ))=0
When the product of factors equals 0,at least one factor is 0
r2=0(cos4(θ)−sin4(θ))r2−21sin(2θ)=0
Evaluate
r=0(cos4(θ)−sin4(θ))r2−21sin(2θ)=0
Solution
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Factor the expression
(cos4(θ)−sin4(θ))r2−21sin(2θ)=0
Subtract the terms
(cos4(θ)−sin4(θ))r2−21sin(2θ)−(−21sin(2θ))=0−(−21sin(2θ))
Evaluate
(cos4(θ)−sin4(θ))r2=21sin(2θ)
Divide the terms
r2=2cos4(θ)−2sin4(θ)sin(2θ)
Simplify the expression
r2=2tan(2θ)
Evaluate the power
r=±2tan(2θ)
Simplify the expression
More Steps

Evaluate
2tan(2θ)
To take a root of a fraction,take the root of the numerator and denominator separately
2tan(2θ)
Multiply by the Conjugate
2×2tan(2θ)×2
Calculate
2tan(2θ)×2
Calculate
22tan(2θ)
r=±22tan(2θ)
Separate into possible cases
r=22tan(2θ)r=−22tan(2θ)
r=0r=22tan(2θ)r=−22tan(2θ)
Show Solution

Find the first derivative
Find the derivative with respect to x
Find the derivative with respect to y
dxdy=4y3+x−y+4x3
Calculate
x4−y4=xy
Take the derivative of both sides
dxd(x4−y4)=dxd(xy)
Calculate the derivative
More Steps

Evaluate
dxd(x4−y4)
Use differentiation rules
dxd(x4)+dxd(−y4)
Use dxdxn=nxn−1 to find derivative
4x3+dxd(−y4)
Evaluate the derivative
More Steps

Evaluate
dxd(−y4)
Use differentiation rules
dyd(−y4)×dxdy
Evaluate the derivative
−4y3dxdy
4x3−4y3dxdy
4x3−4y3dxdy=dxd(xy)
Calculate the derivative
More Steps

Evaluate
dxd(xy)
Use differentiation rules
dxd(x)×y+x×dxd(y)
Use dxdxn=nxn−1 to find derivative
y+x×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
y+xdxdy
4x3−4y3dxdy=y+xdxdy
Move the expression to the left side
4x3−4y3dxdy−xdxdy=y
Move the expression to the right side
−4y3dxdy−xdxdy=y−4x3
Collect like terms by calculating the sum or difference of their coefficients
(−4y3−x)dxdy=y−4x3
Divide both sides
−4y3−x(−4y3−x)dxdy=−4y3−xy−4x3
Divide the numbers
dxdy=−4y3−xy−4x3
Solution
More Steps

Evaluate
−4y3−xy−4x3
Use b−a=−ba=−ba to rewrite the fraction
−4y3+xy−4x3
Rewrite the expression
4y3+x−y+4x3
dxdy=4y3+x−y+4x3
Show Solution

Find the second derivative
Find the second derivative with respect to x
Find the second derivative with respect to y
dx2d2y=64y9+48y6x+12y3x2+x3−4y4+160x3y3+2xy+4x4+192y6x2−192x6y2
Calculate
x4−y4=xy
Take the derivative of both sides
dxd(x4−y4)=dxd(xy)
Calculate the derivative
More Steps

Evaluate
dxd(x4−y4)
Use differentiation rules
dxd(x4)+dxd(−y4)
Use dxdxn=nxn−1 to find derivative
4x3+dxd(−y4)
Evaluate the derivative
More Steps

Evaluate
dxd(−y4)
Use differentiation rules
dyd(−y4)×dxdy
Evaluate the derivative
−4y3dxdy
4x3−4y3dxdy
4x3−4y3dxdy=dxd(xy)
Calculate the derivative
More Steps

Evaluate
dxd(xy)
Use differentiation rules
dxd(x)×y+x×dxd(y)
Use dxdxn=nxn−1 to find derivative
y+x×dxd(y)
Evaluate the derivative
More Steps

Evaluate
dxd(y)
Use differentiation rules
dyd(y)×dxdy
Use dxdxn=nxn−1 to find derivative
dxdy
y+xdxdy
4x3−4y3dxdy=y+xdxdy
Move the expression to the left side
4x3−4y3dxdy−xdxdy=y
Move the expression to the right side
−4y3dxdy−xdxdy=y−4x3
Collect like terms by calculating the sum or difference of their coefficients
(−4y3−x)dxdy=y−4x3
Divide both sides
−4y3−x(−4y3−x)dxdy=−4y3−xy−4x3
Divide the numbers
dxdy=−4y3−xy−4x3
Divide the numbers
More Steps

Evaluate
−4y3−xy−4x3
Use b−a=−ba=−ba to rewrite the fraction
−4y3+xy−4x3
Rewrite the expression
4y3+x−y+4x3
dxdy=4y3+x−y+4x3
Take the derivative of both sides
dxd(dxdy)=dxd(4y3+x−y+4x3)
Calculate the derivative
dx2d2y=dxd(4y3+x−y+4x3)
Use differentiation rules
dx2d2y=(4y3+x)2dxd(−y+4x3)×(4y3+x)−(−y+4x3)×dxd(4y3+x)
Calculate the derivative
More Steps

Evaluate
dxd(−y+4x3)
Use differentiation rules
dxd(−y)+dxd(4x3)
Evaluate the derivative
−dxdy+dxd(4x3)
Evaluate the derivative
−dxdy+12x2
dx2d2y=(4y3+x)2(−dxdy+12x2)(4y3+x)−(−y+4x3)×dxd(4y3+x)
Calculate the derivative
More Steps

Evaluate
dxd(4y3+x)
Use differentiation rules
dxd(4y3)+dxd(x)
Evaluate the derivative
12y2dxdy+dxd(x)
Use dxdxn=nxn−1 to find derivative
12y2dxdy+1
dx2d2y=(4y3+x)2(−dxdy+12x2)(4y3+x)−(−y+4x3)(12y2dxdy+1)
Calculate
More Steps

Evaluate
(−dxdy+12x2)(4y3+x)
Use the the distributive property to expand the expression
−dxdy×(4y3+x)+12x2(4y3+x)
Multiply the terms
−4y3dxdy−xdxdy+12x2(4y3+x)
Multiply the terms
−4y3dxdy−xdxdy+48x2y3+12x3
dx2d2y=(4y3+x)2−4y3dxdy−xdxdy+48x2y3+12x3−(−y+4x3)(12y2dxdy+1)
Calculate
More Steps

Evaluate
(−y+4x3)(12y2dxdy+1)
Use the the distributive property to expand the expression
(−y+4x3)×12y2dxdy+(−y+4x3)×1
Multiply the terms
−12y3dxdy+48x3y2dxdy+(−y+4x3)×1
Any expression multiplied by 1 remains the same
−12y3dxdy+48x3y2dxdy−y+4x3
dx2d2y=(4y3+x)2−4y3dxdy−xdxdy+48x2y3+12x3−(−12y3dxdy+48x3y2dxdy−y+4x3)
Calculate
More Steps

Calculate
−4y3dxdy−xdxdy+48x2y3+12x3−(−12y3dxdy+48x3y2dxdy−y+4x3)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
−4y3dxdy−xdxdy+48x2y3+12x3+12y3dxdy−48x3y2dxdy+y−4x3
Add the terms
8y3dxdy−xdxdy+48x2y3+12x3−48x3y2dxdy+y−4x3
Subtract the terms
8y3dxdy−xdxdy+48x2y3+8x3−48x3y2dxdy+y
dx2d2y=(4y3+x)28y3dxdy−xdxdy+48x2y3+8x3−48x3y2dxdy+y
Use equation dxdy=4y3+x−y+4x3 to substitute
dx2d2y=(4y3+x)28y3×4y3+x−y+4x3−x×4y3+x−y+4x3+48x2y3+8x3−48x3y2×4y3+x−y+4x3+y
Solution
More Steps

Calculate
(4y3+x)28y3×4y3+x−y+4x3−x×4y3+x−y+4x3+48x2y3+8x3−48x3y2×4y3+x−y+4x3+y
Multiply the terms
(4y3+x)24y3+x8y3(−y+4x3)−x×4y3+x−y+4x3+48x2y3+8x3−48x3y2×4y3+x−y+4x3+y
Multiply the terms
(4y3+x)24y3+x8y3(−y+4x3)−4y3+xx(−y+4x3)+48x2y3+8x3−48x3y2×4y3+x−y+4x3+y
Multiply the terms
(4y3+x)24y3+x8y3(−y+4x3)−4y3+xx(−y+4x3)+48x2y3+8x3−4y3+x48x3y2(−y+4x3)+y
Calculate the sum or difference
More Steps

Evaluate
4y3+x8y3(−y+4x3)−4y3+xx(−y+4x3)+48x2y3+8x3−4y3+x48x3y2(−y+4x3)+y
Reduce fractions to a common denominator
4y3+x8y3(−y+4x3)−4y3+xx(−y+4x3)+4y3+x48x2y3(4y3+x)+4y3+x8x3(4y3+x)−4y3+x48x3y2(−y+4x3)+4y3+xy(4y3+x)
Write all numerators above the common denominator
4y3+x8y3(−y+4x3)−x(−y+4x3)+48x2y3(4y3+x)+8x3(4y3+x)−48x3y2(−y+4x3)+y(4y3+x)
Multiply the terms
4y3+x−8y4+32x3y3−x(−y+4x3)+48x2y3(4y3+x)+8x3(4y3+x)−48x3y2(−y+4x3)+y(4y3+x)
Multiply the terms
4y3+x−8y4+32x3y3−(−xy+4x4)+48x2y3(4y3+x)+8x3(4y3+x)−48x3y2(−y+4x3)+y(4y3+x)
Multiply the terms
4y3+x−8y4+32x3y3−(−xy+4x4)+192y6x2+48x3y3+8x3(4y3+x)−48x3y2(−y+4x3)+y(4y3+x)
Multiply the terms
4y3+x−8y4+32x3y3−(−xy+4x4)+192y6x2+48x3y3+32y3x3+8x4−48x3y2(−y+4x3)+y(4y3+x)
Multiply the terms
4y3+x−8y4+32x3y3−(−xy+4x4)+192y6x2+48x3y3+32y3x3+8x4−(−48y3x3+192x6y2)+y(4y3+x)
Multiply the terms
4y3+x−8y4+32x3y3−(−xy+4x4)+192y6x2+48x3y3+32y3x3+8x4−(−48y3x3+192x6y2)+4y4+yx
Calculate the sum or difference
4y3+x−4y4+160x3y3+2xy+4x4+192y6x2−192x6y2
(4y3+x)24y3+x−4y4+160x3y3+2xy+4x4+192y6x2−192x6y2
Multiply by the reciprocal
4y3+x−4y4+160x3y3+2xy+4x4+192y6x2−192x6y2×(4y3+x)21
Multiply the terms
(4y3+x)(4y3+x)2−4y4+160x3y3+2xy+4x4+192y6x2−192x6y2
Multiply the terms
(4y3+x)3−4y4+160x3y3+2xy+4x4+192y6x2−192x6y2
Expand the expression
More Steps

Evaluate
(4y3+x)3
Use (a+b)3=a3+3a2b+3ab2+b3 to expand the expression
(4y3)3+3(4y3)2x+3×4y3x2+x3
Calculate
64y9+48y6x+12y3x2+x3
64y9+48y6x+12y3x2+x3−4y4+160x3y3+2xy+4x4+192y6x2−192x6y2
dx2d2y=64y9+48y6x+12y3x2+x3−4y4+160x3y3+2xy+4x4+192y6x2−192x6y2
Show Solution
