Question
Solve the equation
Solve for x
Solve for y
x=58y5
Evaluate
8y5=5x
Swap the sides of the equation
5x=8y5
Divide both sides
55x=58y5
Solution
x=58y5
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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
8y5=5x
To test if the graph of 8y5=5x is symmetry with respect to the origin,substitute -x for x and -y for y
8(−y)5=5(−x)
Evaluate
More Steps

Evaluate
8(−y)5
Rewrite the expression
8(−y5)
Multiply the numbers
−8y5
−8y5=5(−x)
Evaluate
−8y5=−5x
Solution
Symmetry with respect to the origin
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Rewrite the equation
r=0r=2342560cos(θ)csc(θ)×∣csc(θ)∣r=−2342560cos(θ)csc(θ)×∣csc(θ)∣
Evaluate
8y5=5x
Move the expression to the left side
8y5−5x=0
To convert the equation to polar coordinates,substitute x for rcos(θ) and y for rsin(θ)
8(sin(θ)×r)5−5cos(θ)×r=0
Factor the expression
8sin5(θ)×r5−5cos(θ)×r=0
Factor the expression
r(8sin5(θ)×r4−5cos(θ))=0
When the product of factors equals 0,at least one factor is 0
r=08sin5(θ)×r4−5cos(θ)=0
Solution
More Steps

Factor the expression
8sin5(θ)×r4−5cos(θ)=0
Subtract the terms
8sin5(θ)×r4−5cos(θ)−(−5cos(θ))=0−(−5cos(θ))
Evaluate
8sin5(θ)×r4=5cos(θ)
Divide the terms
r4=8sin5(θ)5cos(θ)
Simplify the expression
r4=85cos(θ)csc5(θ)
Evaluate the power
r=±485cos(θ)csc5(θ)
Simplify the expression
More Steps

Evaluate
485cos(θ)csc5(θ)
To take a root of a fraction,take the root of the numerator and denominator separately
4845cos(θ)csc5(θ)
Simplify the radical expression
4845cos(θ)csc(θ)×∣csc(θ)∣
Multiply by the Conjugate
48×48345cos(θ)csc(θ)×∣csc(θ)∣×483
Calculate
2345cos(θ)csc(θ)×∣csc(θ)∣×483
Calculate the product
2342560cos(θ)csc(θ)×∣csc(θ)∣
r=±2342560cos(θ)csc(θ)×∣csc(θ)∣
Separate into possible cases
r=2342560cos(θ)csc(θ)×∣csc(θ)∣r=−2342560cos(θ)csc(θ)×∣csc(θ)∣
r=0r=2342560cos(θ)csc(θ)×∣csc(θ)∣r=−2342560cos(θ)csc(θ)×∣csc(θ)∣
Show Solution

Find the first derivative
Find the derivative with respect to x
Find the derivative with respect to y
dxdy=8y41
Calculate
8y5=5x
Take the derivative of both sides
dxd(8y5)=dxd(5x)
Calculate the derivative
More Steps

Evaluate
dxd(8y5)
Use differentiation rules
dyd(8y5)×dxdy
Evaluate the derivative
More Steps

Evaluate
dyd(8y5)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
8×dyd(y5)
Use dxdxn=nxn−1 to find derivative
8×5y4
Multiply the terms
40y4
40y4dxdy
40y4dxdy=dxd(5x)
Calculate the derivative
More Steps

Evaluate
dxd(5x)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
5×dxd(x)
Use dxdxn=nxn−1 to find derivative
5×1
Any expression multiplied by 1 remains the same
5
40y4dxdy=5
Divide both sides
40y440y4dxdy=40y45
Divide the numbers
dxdy=40y45
Solution
dxdy=8y41
Show Solution

Find the second derivative
Find the second derivative with respect to x
Find the second derivative with respect to y
dx2d2y=−16y91
Calculate
8y5=5x
Take the derivative of both sides
dxd(8y5)=dxd(5x)
Calculate the derivative
More Steps

Evaluate
dxd(8y5)
Use differentiation rules
dyd(8y5)×dxdy
Evaluate the derivative
More Steps

Evaluate
dyd(8y5)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
8×dyd(y5)
Use dxdxn=nxn−1 to find derivative
8×5y4
Multiply the terms
40y4
40y4dxdy
40y4dxdy=dxd(5x)
Calculate the derivative
More Steps

Evaluate
dxd(5x)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
5×dxd(x)
Use dxdxn=nxn−1 to find derivative
5×1
Any expression multiplied by 1 remains the same
5
40y4dxdy=5
Divide both sides
40y440y4dxdy=40y45
Divide the numbers
dxdy=40y45
Cancel out the common factor 5
dxdy=8y41
Take the derivative of both sides
dxd(dxdy)=dxd(8y41)
Calculate the derivative
dx2d2y=dxd(8y41)
Use differentiation rules
dx2d2y=81×dxd(y41)
Rewrite the expression in exponential form
dx2d2y=81×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=81(−4y−5dxdy)
Rewrite the expression
dx2d2y=81(−y54dxdy)
Calculate
dx2d2y=−2y5dxdy
Use equation dxdy=8y41 to substitute
dx2d2y=−2y58y41
Solution
More Steps

Calculate
−2y58y41
Divide the terms
More Steps

Evaluate
2y58y41
Multiply by the reciprocal
8y41×2y51
Multiply the terms
8y4×2y51
Multiply the terms
16y91
−16y91
dx2d2y=−16y91
Show Solution
