Question
Solve the equation
Solve for x
Solve for y
x=2y5
Evaluate
2y5=x
Solution
x=2y5
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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
2y5=x
To test if the graph of 2y5=x is symmetry with respect to the origin,substitute -x for x and -y for y
2(−y)5=−x
Evaluate
More Steps

Evaluate
2(−y)5
Rewrite the expression
2(−y5)
Multiply the numbers
−2y5
−2y5=−x
Solution
Symmetry with respect to the origin
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Rewrite the equation
r=0r=42sin5(θ)cos(θ)r=−42sin5(θ)cos(θ)
Evaluate
2(y5)=x
Evaluate
2y5=x
Move the expression to the left side
2y5−x=0
To convert the equation to polar coordinates,substitute x for rcos(θ) and y for rsin(θ)
2(sin(θ)×r)5−cos(θ)×r=0
Factor the expression
2sin5(θ)×r5−cos(θ)×r=0
Factor the expression
r(2sin5(θ)×r4−cos(θ))=0
When the product of factors equals 0,at least one factor is 0
r=02sin5(θ)×r4−cos(θ)=0
Solution
More Steps

Factor the expression
2sin5(θ)×r4−cos(θ)=0
Subtract the terms
2sin5(θ)×r4−cos(θ)−(−cos(θ))=0−(−cos(θ))
Evaluate
2sin5(θ)×r4=cos(θ)
Divide the terms
r4=2sin5(θ)cos(θ)
Evaluate the power
r=±42sin5(θ)cos(θ)
Separate into possible cases
r=42sin5(θ)cos(θ)r=−42sin5(θ)cos(θ)
r=0r=42sin5(θ)cos(θ)r=−42sin5(θ)cos(θ)
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Find the first derivative
Find the derivative with respect to x
Find the derivative with respect to y
dxdy=10y41
Calculate
2(y5)=x
Simplify the expression
2y5=x
Take the derivative of both sides
dxd(2y5)=dxd(x)
Calculate the derivative
More Steps

Evaluate
dxd(2y5)
Use differentiation rules
dyd(2y5)×dxdy
Evaluate the derivative
More Steps

Evaluate
dyd(2y5)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
2×dyd(y5)
Use dxdxn=nxn−1 to find derivative
2×5y4
Multiply the terms
10y4
10y4dxdy
10y4dxdy=dxd(x)
Use dxdxn=nxn−1 to find derivative
10y4dxdy=1
Divide both sides
10y410y4dxdy=10y41
Solution
dxdy=10y41
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Find the second derivative
Find the second derivative with respect to x
Find the second derivative with respect to y
dx2d2y=−25y91
Calculate
2(y5)=x
Simplify the expression
2y5=x
Take the derivative of both sides
dxd(2y5)=dxd(x)
Calculate the derivative
More Steps

Evaluate
dxd(2y5)
Use differentiation rules
dyd(2y5)×dxdy
Evaluate the derivative
More Steps

Evaluate
dyd(2y5)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
2×dyd(y5)
Use dxdxn=nxn−1 to find derivative
2×5y4
Multiply the terms
10y4
10y4dxdy
10y4dxdy=dxd(x)
Use dxdxn=nxn−1 to find derivative
10y4dxdy=1
Divide both sides
10y410y4dxdy=10y41
Divide the numbers
dxdy=10y41
Take the derivative of both sides
dxd(dxdy)=dxd(10y41)
Calculate the derivative
dx2d2y=dxd(10y41)
Use differentiation rules
dx2d2y=101×dxd(y41)
Rewrite the expression in exponential form
dx2d2y=101×dxd(y−4)
Calculate the derivative
More Steps

Evaluate
dxd(y−4)
Use differentiation rules
dyd(y−4)×dxdy
Use dxdxn=nxn−1 to find derivative
−4y−5dxdy
dx2d2y=101(−4y−5dxdy)
Rewrite the expression
dx2d2y=101(−y54dxdy)
Calculate
dx2d2y=−5y52dxdy
Use equation dxdy=10y41 to substitute
dx2d2y=−5y52×10y41
Solution
More Steps

Calculate
−5y52×10y41
Multiply the terms
More Steps

Multiply the terms
2×10y41
Cancel out the common factor 2
1×5y41
Multiply the terms
5y41
−5y55y41
Divide the terms
More Steps

Evaluate
5y55y41
Multiply by the reciprocal
5y41×5y51
Multiply the terms
5y4×5y51
Multiply the terms
25y91
−25y91
dx2d2y=−25y91
Show Solution
