Question
Solve the equation
Solve for x
Solve for y
x=0
Evaluate
−3x6y=0
Rewrite the expression
−3yx6=0
Rewrite the expression
x6=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
−3x6y=0
To test if the graph of −3x6y=0 is symmetry with respect to the origin,substitute -x for x and -y for y
−3(−x)6(−y)=0
Evaluate
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Evaluate
−3(−x)6(−y)
Any expression multiplied by 1 remains the same
3(−x)6y
Multiply the terms
3x6y
3x6y=0
Solution
Symmetry with respect to the origin
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Rewrite the equation
r=0θ=2kπ,k∈Z
Evaluate
−3x6y=0
To convert the equation to polar coordinates,substitute x for rcos(θ) and y for rsin(θ)
−3(cos(θ)×r)6sin(θ)×r=0
Factor the expression
−3cos6(θ)sin(θ)×r7=0
Separate into possible cases
r7=0−3cos6(θ)sin(θ)=0
Evaluate
r=0−3cos6(θ)sin(θ)=0
Solution
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Evaluate
−3cos6(θ)sin(θ)=0
Elimination the left coefficient
cos6(θ)sin(θ)=0
Separate the equation into 2 possible cases
cos6(θ)=0sin(θ)=0
Solve the equation
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Evaluate
cos6(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
Show Solution

Find the first derivative
Find the derivative with respect to x
Find the derivative with respect to y
dxdy=−x6y
Calculate
−3x6y=0
Take the derivative of both sides
dxd(−3x6y)=dxd(0)
Calculate the derivative
More Steps

Evaluate
dxd(−3x6y)
Use differentiation rules
dxd(−3x6)×y−3x6×dxd(y)
Evaluate the derivative
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Evaluate
dxd(−3x6)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
−3×dxd(x6)
Use dxdxn=nxn−1 to find derivative
−3×6x5
Multiply the terms
−18x5
−18x5y−3x6×dxd(y)
Evaluate the derivative
More Steps

Evaluate
dxd(y)
Use differentiation rules
dyd(y)×dxdy
Use dxdxn=nxn−1 to find derivative
dxdy
−18x5y−3x6dxdy
−18x5y−3x6dxdy=dxd(0)
Calculate the derivative
−18x5y−3x6dxdy=0
Move the expression to the right-hand side and change its sign
−3x6dxdy=0+18x5y
Add the terms
−3x6dxdy=18x5y
Divide both sides
−3x6−3x6dxdy=−3x618x5y
Divide the numbers
dxdy=−3x618x5y
Solution
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Evaluate
−3x618x5y
Cancel out the common factor 3
−x66x5y
Reduce the fraction
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Evaluate
x6x5
Use the product rule aman=an−m to simplify the expression
x6−51
Subtract the terms
x11
Simplify
x1
−x6y
Use b−a=−ba=−ba to rewrite the fraction
−x6y
dxdy=−x6y
Show Solution

Find the second derivative
Find the second derivative with respect to x
Find the second derivative with respect to y
dx2d2y=x242y
Calculate
−3x6y=0
Take the derivative of both sides
dxd(−3x6y)=dxd(0)
Calculate the derivative
More Steps

Evaluate
dxd(−3x6y)
Use differentiation rules
dxd(−3x6)×y−3x6×dxd(y)
Evaluate the derivative
More Steps

Evaluate
dxd(−3x6)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
−3×dxd(x6)
Use dxdxn=nxn−1 to find derivative
−3×6x5
Multiply the terms
−18x5
−18x5y−3x6×dxd(y)
Evaluate the derivative
More Steps

Evaluate
dxd(y)
Use differentiation rules
dyd(y)×dxdy
Use dxdxn=nxn−1 to find derivative
dxdy
−18x5y−3x6dxdy
−18x5y−3x6dxdy=dxd(0)
Calculate the derivative
−18x5y−3x6dxdy=0
Move the expression to the right-hand side and change its sign
−3x6dxdy=0+18x5y
Add the terms
−3x6dxdy=18x5y
Divide both sides
−3x6−3x6dxdy=−3x618x5y
Divide the numbers
dxdy=−3x618x5y
Divide the numbers
More Steps

Evaluate
−3x618x5y
Cancel out the common factor 3
−x66x5y
Reduce the fraction
More Steps

Evaluate
x6x5
Use the product rule aman=an−m to simplify the expression
x6−51
Subtract the terms
x11
Simplify
x1
−x6y
Use b−a=−ba=−ba to rewrite the fraction
−x6y
dxdy=−x6y
Take the derivative of both sides
dxd(dxdy)=dxd(−x6y)
Calculate the derivative
dx2d2y=dxd(−x6y)
Use differentiation rules
dx2d2y=−x2dxd(6y)×x−6y×dxd(x)
Calculate the derivative
More Steps

Evaluate
dxd(6y)
Simplify
6×dxd(y)
Calculate
6dxdy
dx2d2y=−x26dxdy×x−6y×dxd(x)
Use dxdxn=nxn−1 to find derivative
dx2d2y=−x26dxdy×x−6y×1
Use the commutative property to reorder the terms
dx2d2y=−x26xdxdy−6y×1
Any expression multiplied by 1 remains the same
dx2d2y=−x26xdxdy−6y
Use equation dxdy=−x6y to substitute
dx2d2y=−x26x(−x6y)−6y
Solution
More Steps

Calculate
−x26x(−x6y)−6y
Multiply
More Steps

Multiply the terms
6x(−x6y)
Any expression multiplied by 1 remains the same
−6x×x6y
Multiply the terms
−36y
−x2−36y−6y
Subtract the terms
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Simplify
−36y−6y
Collect like terms by calculating the sum or difference of their coefficients
(−36−6)y
Subtract the numbers
−42y
−x2−42y
Divide the terms
−(−x242y)
Calculate
x242y
dx2d2y=x242y
Show Solution
