Question
Solve the equation
Solve for x
Solve for y
x=5y3225y2
Evaluate
5x3y=9
Rewrite the expression
5yx3=9
Divide both sides
5y5yx3=5y9
Divide the numbers
x3=5y9
Take the 3-th root on both sides of the equation
3x3=35y9
Calculate
x=35y9
Solution
More Steps

Evaluate
35y9
To take a root of a fraction,take the root of the numerator and denominator separately
35y39
Multiply by the Conjugate
35y×352y239×352y2
Calculate
5y39×352y2
Calculate
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Evaluate
39×352y2
The product of roots with the same index is equal to the root of the product
39×52y2
Calculate the product
3225y2
5y3225y2
x=5y3225y2
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
5x3y=9
To test if the graph of 5x3y=9 is symmetry with respect to the origin,substitute -x for x and -y for y
5(−x)3(−y)=9
Evaluate
More Steps

Evaluate
5(−x)3(−y)
Any expression multiplied by 1 remains the same
−5(−x)3y
Multiply the terms
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Evaluate
5(−x)3
Rewrite the expression
5(−x3)
Multiply the numbers
−5x3
−(−5x3y)
Multiply the first two terms
5x3y
5x3y=9
Solution
Symmetry with respect to the origin
Show Solution

Rewrite the equation
r=45cos3(θ)sin(θ)3r=−45cos3(θ)sin(θ)3
Evaluate
5x3y=9
To convert the equation to polar coordinates,substitute x for rcos(θ) and y for rsin(θ)
5(cos(θ)×r)3sin(θ)×r=9
Factor the expression
5cos3(θ)sin(θ)×r4=9
Divide the terms
r4=5cos3(θ)sin(θ)9
Evaluate the power
r=±45cos3(θ)sin(θ)9
Simplify the expression
More Steps

Evaluate
45cos3(θ)sin(θ)9
To take a root of a fraction,take the root of the numerator and denominator separately
45cos3(θ)sin(θ)49
Simplify the radical expression
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Evaluate
49
Write the number in exponential form with the base of 3
432
Reduce the index of the radical and exponent with 2
3
45cos3(θ)sin(θ)3
r=±45cos3(θ)sin(θ)3
Solution
r=45cos3(θ)sin(θ)3r=−45cos3(θ)sin(θ)3
Show Solution

Find the first derivative
Find the derivative with respect to x
Find the derivative with respect to y
dxdy=−x3y
Calculate
5x3y=9
Take the derivative of both sides
dxd(5x3y)=dxd(9)
Calculate the derivative
More Steps

Evaluate
dxd(5x3y)
Use differentiation rules
dxd(5x3)×y+5x3×dxd(y)
Evaluate the derivative
More Steps

Evaluate
dxd(5x3)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
5×dxd(x3)
Use dxdxn=nxn−1 to find derivative
5×3x2
Multiply the terms
15x2
15x2y+5x3×dxd(y)
Evaluate the derivative
More Steps

Evaluate
dxd(y)
Use differentiation rules
dyd(y)×dxdy
Use dxdxn=nxn−1 to find derivative
dxdy
15x2y+5x3dxdy
15x2y+5x3dxdy=dxd(9)
Calculate the derivative
15x2y+5x3dxdy=0
Move the expression to the right-hand side and change its sign
5x3dxdy=0−15x2y
Removing 0 doesn't change the value,so remove it from the expression
5x3dxdy=−15x2y
Divide both sides
5x35x3dxdy=5x3−15x2y
Divide the numbers
dxdy=5x3−15x2y
Solution
More Steps

Evaluate
5x3−15x2y
Cancel out the common factor 5
x3−3x2y
Reduce the fraction
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Evaluate
x3x2
Use the product rule aman=an−m to simplify the expression
x3−21
Subtract the terms
x11
Simplify
x1
x−3y
Use b−a=−ba=−ba to rewrite the fraction
−x3y
dxdy=−x3y
Show Solution

Find the second derivative
Find the second derivative with respect to x
Find the second derivative with respect to y
dx2d2y=x212y
Calculate
5x3y=9
Take the derivative of both sides
dxd(5x3y)=dxd(9)
Calculate the derivative
More Steps

Evaluate
dxd(5x3y)
Use differentiation rules
dxd(5x3)×y+5x3×dxd(y)
Evaluate the derivative
More Steps

Evaluate
dxd(5x3)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
5×dxd(x3)
Use dxdxn=nxn−1 to find derivative
5×3x2
Multiply the terms
15x2
15x2y+5x3×dxd(y)
Evaluate the derivative
More Steps

Evaluate
dxd(y)
Use differentiation rules
dyd(y)×dxdy
Use dxdxn=nxn−1 to find derivative
dxdy
15x2y+5x3dxdy
15x2y+5x3dxdy=dxd(9)
Calculate the derivative
15x2y+5x3dxdy=0
Move the expression to the right-hand side and change its sign
5x3dxdy=0−15x2y
Removing 0 doesn't change the value,so remove it from the expression
5x3dxdy=−15x2y
Divide both sides
5x35x3dxdy=5x3−15x2y
Divide the numbers
dxdy=5x3−15x2y
Divide the numbers
More Steps

Evaluate
5x3−15x2y
Cancel out the common factor 5
x3−3x2y
Reduce the fraction
More Steps

Evaluate
x3x2
Use the product rule aman=an−m to simplify the expression
x3−21
Subtract the terms
x11
Simplify
x1
x−3y
Use b−a=−ba=−ba to rewrite the fraction
−x3y
dxdy=−x3y
Take the derivative of both sides
dxd(dxdy)=dxd(−x3y)
Calculate the derivative
dx2d2y=dxd(−x3y)
Use differentiation rules
dx2d2y=−x2dxd(3y)×x−3y×dxd(x)
Calculate the derivative
More Steps

Evaluate
dxd(3y)
Simplify
3×dxd(y)
Calculate
3dxdy
dx2d2y=−x23dxdy×x−3y×dxd(x)
Use dxdxn=nxn−1 to find derivative
dx2d2y=−x23dxdy×x−3y×1
Use the commutative property to reorder the terms
dx2d2y=−x23xdxdy−3y×1
Any expression multiplied by 1 remains the same
dx2d2y=−x23xdxdy−3y
Use equation dxdy=−x3y to substitute
dx2d2y=−x23x(−x3y)−3y
Solution
More Steps

Calculate
−x23x(−x3y)−3y
Multiply
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Multiply the terms
3x(−x3y)
Any expression multiplied by 1 remains the same
−3x×x3y
Multiply the terms
−9y
−x2−9y−3y
Subtract the terms
More Steps

Simplify
−9y−3y
Collect like terms by calculating the sum or difference of their coefficients
(−9−3)y
Subtract the numbers
−12y
−x2−12y
Divide the terms
−(−x212y)
Calculate
x212y
dx2d2y=x212y
Show Solution
