Question
Solve the equation
Solve for x
Solve for y
x=arcsin(∣y∣y)
Evaluate
y=∣y∣×sin(x)
Swap the sides of the equation
∣y∣×sin(x)=y
Multiply both sides of the equation by ∣y∣1
∣y∣×sin(x)×∣y∣1=y×∣y∣1
Calculate
sin(x)=y×∣y∣1
Calculate the product
sin(x)=∣y∣y
Solution
x=arcsin(∣y∣y)
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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
y=∣y∣×sin(x)
To test if the graph of y=∣y∣×sin(x) is symmetry with respect to the origin,substitute -x for x and -y for y
−y=∣−y∣×sin(−x)
Evaluate
More Steps

Evaluate
∣−y∣×sin(−x)
Calculate the absolute value
∣y∣×sin(−x)
Use sin(−t)=−sin(t) to transform the expression
∣y∣×(−sin(x))
Calculate
−∣y∣×sin(x)
−y=−∣y∣×sin(x)
Solution
Symmetry with respect to the origin
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Find the first derivative
Find the derivative with respect to x
Find the derivative with respect to y
dxdy=∣y∣−ysin(x)y2cos(x)
Calculate
y=∣y∣⋅sin(x)
Take the derivative of both sides
dxd(y)=dxd(∣y∣×sin(x))
Calculate the derivative
More Steps

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

Evaluate
dxd(∣y∣×sin(x))
Use differentiation rules
dxd(∣y∣)×sin(x)+∣y∣×dxd(sin(x))
Evaluate the derivative
More Steps

Evaluate
dxd(∣y∣)
Rewrite the expression
dxd(y2)
Use differentiation rules
21×∣y∣dxd(y2)
Calculate
∣y∣ydxdy
∣y∣ydxdy×sin(x)+∣y∣×dxd(sin(x))
Use dxd(sinx)=cosx to find derivative
∣y∣ydxdy×sin(x)+∣y∣×cos(x)
dxdy=∣y∣ydxdy×sin(x)+∣y∣×cos(x)
Rewrite the expression
dxdy=∣y∣ysin(x)dxdy+∣y∣×cos(x)
Multiply both sides of the equation by LCD
dxdy∣y∣=(∣y∣ysin(x)dxdy+∣y∣×cos(x))∣y∣
Simplify the equation
∣y∣×dxdy=(∣y∣ysin(x)dxdy+∣y∣×cos(x))∣y∣
Simplify the equation
More Steps

Evaluate
(∣y∣ysin(x)dxdy+∣y∣×cos(x))∣y∣
Apply the distributive property
∣y∣ysin(x)dxdy∣y∣+∣y∣×cos(x)×∣y∣
Simplify
ysin(x)dxdy+∣y∣×cos(x)×∣y∣
Multiply the terms
ysin(x)dxdy+y2cos(x)
∣y∣×dxdy=ysin(x)dxdy+y2cos(x)
Move the variable to the left side
∣y∣×dxdy−ysin(x)dxdy=y2cos(x)
Collect like terms by calculating the sum or difference of their coefficients
(∣y∣−ysin(x))dxdy=y2cos(x)
Divide both sides
∣y∣−ysin(x)(∣y∣−ysin(x))dxdy=∣y∣−ysin(x)y2cos(x)
Solution
dxdy=∣y∣−ysin(x)y2cos(x)
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