Question
Solve the equation
Solve for x
Solve for y
x=0
Evaluate
xy5=0
Rewrite the expression
y5x=0
Solution
x=0
Show Solution

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
xy5=0
To test if the graph of xy5=0 is symmetry with respect to the origin,substitute -x for x and -y for y
−x(−y)5=0
Evaluate
More Steps

Evaluate
−x(−y)5
Rewrite the expression
−x(−y5)
Multiplying or dividing an even number of negative terms equals a positive
xy5
xy5=0
Solution
Symmetry with respect to the origin
Show Solution

Rewrite the equation
r=0θ=2kπ,k∈Z
Evaluate
xy5=0
To convert the equation to polar coordinates,substitute x for rcos(θ) and y for rsin(θ)
cos(θ)×r(sin(θ)×r)5=0
Factor the expression
cos(θ)sin5(θ)×r6=0
Simplify the expression
sin5(θ)cos(θ)×r6=0
Separate into possible cases
r6=0sin5(θ)cos(θ)=0
Evaluate
r=0sin5(θ)cos(θ)=0
Solution
More Steps

Evaluate
sin5(θ)cos(θ)=0
Separate the equation into 2 possible cases
sin5(θ)=0cos(θ)=0
Solve the equation
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Evaluate
sin5(θ)=0
The only way a power can be 0 is when the base equals 0
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=kπ,k∈Zcos(θ)=0
Solve the equation
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Evaluate
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=kπ,k∈Zθ=2π+kπ,k∈Z
Find the union
θ=2kπ,k∈Z
r=0θ=2kπ,k∈Z
Show Solution

Find the first derivative
Find the derivative with respect to x
Find the derivative with respect to y
dxdy=−5xy
Calculate
xy5=0
Take the derivative of both sides
dxd(xy5)=dxd(0)
Calculate the derivative
More Steps

Evaluate
dxd(xy5)
Use differentiation rules
dxd(x)×y5+x×dxd(y5)
Use dxdxn=nxn−1 to find derivative
y5+x×dxd(y5)
Evaluate the derivative
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Evaluate
dxd(y5)
Use differentiation rules
dyd(y5)×dxdy
Use dxdxn=nxn−1 to find derivative
5y4dxdy
y5+5xy4dxdy
y5+5xy4dxdy=dxd(0)
Calculate the derivative
y5+5xy4dxdy=0
Move the expression to the right-hand side and change its sign
5xy4dxdy=0−y5
Removing 0 doesn't change the value,so remove it from the expression
5xy4dxdy=−y5
Divide both sides
5xy45xy4dxdy=5xy4−y5
Divide the numbers
dxdy=5xy4−y5
Solution
More Steps

Evaluate
5xy4−y5
Rewrite the expression
5xy4y4(−y)
Reduce the fraction
5x−y
Use b−a=−ba=−ba to rewrite the fraction
−5xy
dxdy=−5xy
Show Solution

Find the second derivative
Find the second derivative with respect to x
Find the second derivative with respect to y
dx2d2y=25x26y
Calculate
xy5=0
Take the derivative of both sides
dxd(xy5)=dxd(0)
Calculate the derivative
More Steps

Evaluate
dxd(xy5)
Use differentiation rules
dxd(x)×y5+x×dxd(y5)
Use dxdxn=nxn−1 to find derivative
y5+x×dxd(y5)
Evaluate the derivative
More Steps

Evaluate
dxd(y5)
Use differentiation rules
dyd(y5)×dxdy
Use dxdxn=nxn−1 to find derivative
5y4dxdy
y5+5xy4dxdy
y5+5xy4dxdy=dxd(0)
Calculate the derivative
y5+5xy4dxdy=0
Move the expression to the right-hand side and change its sign
5xy4dxdy=0−y5
Removing 0 doesn't change the value,so remove it from the expression
5xy4dxdy=−y5
Divide both sides
5xy45xy4dxdy=5xy4−y5
Divide the numbers
dxdy=5xy4−y5
Divide the numbers
More Steps

Evaluate
5xy4−y5
Rewrite the expression
5xy4y4(−y)
Reduce the fraction
5x−y
Use b−a=−ba=−ba to rewrite the fraction
−5xy
dxdy=−5xy
Take the derivative of both sides
dxd(dxdy)=dxd(−5xy)
Calculate the derivative
dx2d2y=dxd(−5xy)
Use differentiation rules
dx2d2y=−(5x)2dxd(y)×5x−y×dxd(5x)
Calculate the derivative
More Steps

Evaluate
dxd(y)
Use differentiation rules
dyd(y)×dxdy
Use dxdxn=nxn−1 to find derivative
dxdy
dx2d2y=−(5x)2dxdy×5x−y×dxd(5x)
Calculate the derivative
More Steps

Evaluate
dxd(5x)
Simplify
5×dxd(x)
Rewrite the expression
5×1
Any expression multiplied by 1 remains the same
5
dx2d2y=−(5x)2dxdy×5x−y×5
Use the commutative property to reorder the terms
dx2d2y=−(5x)25dxdy×x−y×5
Use the commutative property to reorder the terms
dx2d2y=−(5x)25dxdy×x−5y
Use the commutative property to reorder the terms
dx2d2y=−(5x)25xdxdy−5y
Calculate
More Steps

Evaluate
(5x)2
Evaluate the power
52x2
Evaluate the power
25x2
dx2d2y=−25x25xdxdy−5y
Calculate
dx2d2y=−5x2xdxdy−y
Use equation dxdy=−5xy to substitute
dx2d2y=−5x2x(−5xy)−y
Solution
More Steps

Calculate
−5x2x(−5xy)−y
Multiply the terms
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Evaluate
x(−5xy)
Multiplying or dividing an odd number of negative terms equals a negative
−x×5xy
Cancel out the common factor x
−1×5y
Multiply the terms
−5y
−5x2−5y−y
Subtract the terms
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Simplify
−5y−y
Reduce fractions to a common denominator
−5y−5y×5
Write all numerators above the common denominator
5−y−y×5
Use the commutative property to reorder the terms
5−y−5y
Subtract the terms
5−6y
Use b−a=−ba=−ba to rewrite the fraction
−56y
−5x2−56y
Divide the terms
More Steps

Evaluate
5x2−56y
Multiply by the reciprocal
−56y×5x21
Multiply the terms
−5×5x26y
Multiply the terms
−25x26y
−(−25x26y)
Calculate
25x26y
dx2d2y=25x26y
Show Solution
