Question
Solve the equation
Solve for x
Solve for y
x=y2−y30
Evaluate
xy(y−1)=30
Rewrite the expression
(y2−y)x=30
Divide both sides
y2−y(y2−y)x=y2−y30
Solution
x=y2−y30
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
Not symmetry with respect to the origin
Evaluate
xy(y−1)=30
To test if the graph of xy(y−1)=30 is symmetry with respect to the origin,substitute -x for x and -y for y
−x(−y)(−y−1)=30
Evaluate
xy(−y−1)=30
Solution
Not symmetry with respect to the origin
Show Solution

Find the first derivative
Find the derivative with respect to x
Find the derivative with respect to y
dxdy=2xy−x−y2+y
Calculate
(x⋅y)(y−1)=30
Simplify the expression
xy(y−1)=30
Take the derivative of both sides
dxd(xy(y−1))=dxd(30)
Calculate the derivative
More Steps

Evaluate
dxd(xy(y−1))
Use differentiation rules
dxd(xy)×(y−1)+xy×dxd(y−1)
Evaluate 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
y+xdxdy
y2−y+xydxdy−xdxdy+xy×dxd(y−1)
Evaluate the derivative
More Steps

Evaluate
dxd(y−1)
Use differentiation rules
dxd(y)+dxd(−1)
Evaluate the derivative
dxdy+dxd(−1)
Use dxd(c)=0 to find derivative
dxdy+0
Evaluate
dxdy
y2−y+xydxdy−xdxdy+xydxdy
y2−y+xydxdy−xdxdy+xydxdy=dxd(30)
Calculate the derivative
y2−y+xydxdy−xdxdy+xydxdy=0
Calculate the sum or difference
More Steps

Evaluate
xydxdy−xdxdy+xydxdy
Collect like terms by calculating the sum or difference of their coefficients
(xy−x+xy)dxdy
Add the terms
More Steps

Evaluate
xy+xy
Collect like terms by calculating the sum or difference of their coefficients
(1+1)xy
Add the numbers
2xy
(2xy−x)dxdy
y2−y+(2xy−x)dxdy=0
Move the constant to the right side
(2xy−x)dxdy=0−(y2−y)
Subtract the terms
More Steps

Evaluate
0−(y2−y)
Removing 0 doesn't change the value,so remove it from the expression
−(y2−y)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
−y2+y
(2xy−x)dxdy=−y2+y
Divide both sides
2xy−x(2xy−x)dxdy=2xy−x−y2+y
Solution
dxdy=2xy−x−y2+y
Show Solution

Find the second derivative
Find the second derivative with respect to x
Find the second derivative with respect to y
dx2d2y=8y3x2−12y2x2+6yx2−x26y4−12y3+8y2−2y
Calculate
(x⋅y)(y−1)=30
Simplify the expression
xy(y−1)=30
Take the derivative of both sides
dxd(xy(y−1))=dxd(30)
Calculate the derivative
More Steps

Evaluate
dxd(xy(y−1))
Use differentiation rules
dxd(xy)×(y−1)+xy×dxd(y−1)
Evaluate 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
y+xdxdy
y2−y+xydxdy−xdxdy+xy×dxd(y−1)
Evaluate the derivative
More Steps

Evaluate
dxd(y−1)
Use differentiation rules
dxd(y)+dxd(−1)
Evaluate the derivative
dxdy+dxd(−1)
Use dxd(c)=0 to find derivative
dxdy+0
Evaluate
dxdy
y2−y+xydxdy−xdxdy+xydxdy
y2−y+xydxdy−xdxdy+xydxdy=dxd(30)
Calculate the derivative
y2−y+xydxdy−xdxdy+xydxdy=0
Calculate the sum or difference
More Steps

Evaluate
xydxdy−xdxdy+xydxdy
Collect like terms by calculating the sum or difference of their coefficients
(xy−x+xy)dxdy
Add the terms
More Steps

Evaluate
xy+xy
Collect like terms by calculating the sum or difference of their coefficients
(1+1)xy
Add the numbers
2xy
(2xy−x)dxdy
y2−y+(2xy−x)dxdy=0
Move the constant to the right side
(2xy−x)dxdy=0−(y2−y)
Subtract the terms
More Steps

Evaluate
0−(y2−y)
Removing 0 doesn't change the value,so remove it from the expression
−(y2−y)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
−y2+y
(2xy−x)dxdy=−y2+y
Divide both sides
2xy−x(2xy−x)dxdy=2xy−x−y2+y
Divide the numbers
dxdy=2xy−x−y2+y
Take the derivative of both sides
dxd(dxdy)=dxd(2xy−x−y2+y)
Calculate the derivative
dx2d2y=dxd(2xy−x−y2+y)
Use differentiation rules
dx2d2y=(2xy−x)2dxd(−y2+y)×(2xy−x)−(−y2+y)×dxd(2xy−x)
Calculate the derivative
More Steps

Evaluate
dxd(−y2+y)
Use differentiation rules
dxd(−y2)+dxd(y)
Evaluate the derivative
−2ydxdy+dxd(y)
Evaluate the derivative
−2ydxdy+dxdy
dx2d2y=(2xy−x)2(−2ydxdy+dxdy)(2xy−x)−(−y2+y)×dxd(2xy−x)
Calculate the derivative
More Steps

Evaluate
dxd(2xy−x)
Use differentiation rules
dxd(2xy)+dxd(−x)
Evaluate the derivative
2y+2xdxdy+dxd(−x)
Evaluate the derivative
2y+2xdxdy−1
dx2d2y=(2xy−x)2(−2ydxdy+dxdy)(2xy−x)−(−y2+y)(2y+2xdxdy−1)
Calculate
More Steps

Evaluate
(−2ydxdy+dxdy)(2xy−x)
Use the the distributive property to expand the expression
−2ydxdy×(2xy−x)+dxdy×(2xy−x)
Multiply the terms
−4y2xdxdy+2yxdxdy+dxdy×(2xy−x)
Multiply the terms
−4y2xdxdy+2yxdxdy+2xydxdy−xdxdy
Calculate
−4y2xdxdy+4yxdxdy−xdxdy
dx2d2y=(2xy−x)2−4y2xdxdy+4yxdxdy−xdxdy−(−y2+y)(2y+2xdxdy−1)
Calculate
More Steps

Evaluate
(−y2+y)(2y+2xdxdy−1)
Use the the distributive property to expand the expression
(−y2+y)(2y+2xdxdy)+(−y2+y)(−1)
Multiply the terms
−2y3−2y2xdxdy+2y2+2yxdxdy+(−y2+y)(−1)
Multiply the terms
−2y3−2y2xdxdy+2y2+2yxdxdy+y2−y
Calculate
−2y3−2y2xdxdy+3y2+2yxdxdy−y
dx2d2y=(2xy−x)2−4y2xdxdy+4yxdxdy−xdxdy−(−2y3−2y2xdxdy+3y2+2yxdxdy−y)
Calculate
More Steps

Calculate
−4y2xdxdy+4yxdxdy−xdxdy−(−2y3−2y2xdxdy+3y2+2yxdxdy−y)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
−4y2xdxdy+4yxdxdy−xdxdy+2y3+2y2xdxdy−3y2−2yxdxdy+y
Add the terms
−2y2xdxdy+4yxdxdy−xdxdy+2y3−3y2−2yxdxdy+y
Subtract the terms
−2y2xdxdy+2yxdxdy−xdxdy+2y3−3y2+y
dx2d2y=(2xy−x)2−2y2xdxdy+2yxdxdy−xdxdy+2y3−3y2+y
Use equation dxdy=2xy−x−y2+y to substitute
dx2d2y=(2xy−x)2−2y2x×2xy−x−y2+y+2yx×2xy−x−y2+y−x×2xy−x−y2+y+2y3−3y2+y
Solution
More Steps

Calculate
(2xy−x)2−2y2x×2xy−x−y2+y+2yx×2xy−x−y2+y−x×2xy−x−y2+y+2y3−3y2+y
Multiply the terms
(2xy−x)2−2y−12y2(−y2+y)+2yx×2xy−x−y2+y−x×2xy−x−y2+y+2y3−3y2+y
Multiply the terms
More Steps

Multiply the terms
2yx×2xy−x−y2+y
Rewrite the expression
2yx×x(2y−1)−y2+y
Cancel out the common factor x
2y×2y−1−y2+y
Multiply the terms
2y−12y(−y2+y)
(2xy−x)2−2y−12y2(−y2+y)+2y−12y(−y2+y)−x×2xy−x−y2+y+2y3−3y2+y
Multiply the terms
More Steps

Evaluate
−x×2xy−x−y2+y
Rewrite the expression
−x×x(2y−1)−y2+y
Cancel out the common factor x
−1×2y−1−y2+y
Multiply the terms
−2y−1−y2+y
Use b−a=−ba=−ba to rewrite the fraction
2y−1y2−y
(2xy−x)2−2y−12y2(−y2+y)+2y−12y(−y2+y)+2y−1y2−y+2y3−3y2+y
Calculate the sum or difference
More Steps

Evaluate
−2y−12y2(−y2+y)+2y−12y(−y2+y)+2y−1y2−y+2y3−3y2+y
Reduce fractions to a common denominator
−2y−12y2(−y2+y)+2y−12y(−y2+y)+2y−1y2−y+2y−12y3(2y−1)−2y−13y2(2y−1)+2y−1y(2y−1)
Write all numerators above the common denominator
2y−1−2y2(−y2+y)+2y(−y2+y)+y2−y+2y3(2y−1)−3y2(2y−1)+y(2y−1)
Multiply the terms
2y−1−(−2y4+2y3)+2y(−y2+y)+y2−y+2y3(2y−1)−3y2(2y−1)+y(2y−1)
Multiply the terms
2y−1−(−2y4+2y3)−2y3+2y2+y2−y+2y3(2y−1)−3y2(2y−1)+y(2y−1)
Multiply the terms
2y−1−(−2y4+2y3)−2y3+2y2+y2−y+4y4−2y3−3y2(2y−1)+y(2y−1)
Multiply the terms
2y−1−(−2y4+2y3)−2y3+2y2+y2−y+4y4−2y3−(6y3−3y2)+y(2y−1)
Multiply the terms
2y−1−(−2y4+2y3)−2y3+2y2+y2−y+4y4−2y3−(6y3−3y2)+2y2−y
Calculate the sum or difference
2y−16y4−12y3+8y2−2y
(2xy−x)22y−16y4−12y3+8y2−2y
Multiply by the reciprocal
2y−16y4−12y3+8y2−2y×(2xy−x)21
Multiply the terms
(2y−1)(2xy−x)26y4−12y3+8y2−2y
Expand the expression
More Steps

Evaluate
(2y−1)(2xy−x)2
Expand the expression
(2y−1)(4x2y2−4x2y+x2)
Apply the distributive property
2y×4x2y2−2y×4x2y+2yx2−4x2y2−(−4x2y)−x2
Multiply the terms
8y3x2−2y×4x2y+2yx2−4x2y2−(−4x2y)−x2
Multiply the terms
8y3x2−8y2x2+2yx2−4x2y2−(−4x2y)−x2
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
8y3x2−8y2x2+2yx2−4x2y2+4x2y−x2
Subtract the terms
8y3x2−12y2x2+2yx2+4x2y−x2
Add the terms
8y3x2−12y2x2+6yx2−x2
8y3x2−12y2x2+6yx2−x26y4−12y3+8y2−2y
dx2d2y=8y3x2−12y2x2+6yx2−x26y4−12y3+8y2−2y
Show Solution
