Solve x^6/5=y 10/3

Question

Function

Find the x -intercept/zero

Find the y-intercept

x = 0

Evaluate

\frac{x ^{6}}{5} = y \times \frac{10}{3}

To find the x -intercept,set y =0

\frac{x ^{6}}{5} = 0 \times \frac{10}{3}

Any expression multiplied by 0 equals 0

\frac{x ^{6}}{5} = 0

Simplify

x^{6} = 0

Solution

x = 0

Show Solution

Solve the equation

Solve for x

Solve for y

x = \frac{6 12150 y}{3} x = - \frac{6 12150 y}{3}

Evaluate

\frac{x ^{6}}{5} = y \times \frac{10}{3}

Use the commutative property to reorder the terms

\frac{x ^{6}}{5} = \frac{10}{3} y

Rewrite the expression

\frac{x ^{6}}{5} = \frac{10 y}{3}

Multiply both sides of the equation by 5

\frac{x ^{6}}{5} \times 5 = \frac{10 y}{3} \times 5

Multiply the terms

x^{6} = \frac{10 y \times 5}{3}

Divide the terms

x^{6} = \frac{50 y}{3}

Take the root of both sides of the equation and remember to use both positive and negative roots

x = \pm 6 \frac{50 y}{3}

Simplify the expression

More Steps

Evaluate

6 \frac{50 y}{3}

To take a root of a fraction,take the root of the numerator and denominator separately

\frac{6 50 y}{6 3}

Multiply by the Conjugate

\frac{6 50 y \times 6 3 ^{5}}{6 3 \times 6 3 ^{5}}

Calculate

\frac{6 50 y \times 6 3 ^{5}}{3}

Calculate

More Steps

Evaluate

6 50 y \times 6 3^{5}

The product of roots with the same index is equal to the root of the product

6 50 y \times 3^{5}

Calculate the product

6 12150 y

\frac{6 12150 y}{3}

x = \pm \frac{6 12150 y}{3}

Solution

x = \frac{6 12150 y}{3} x = - \frac{6 12150 y}{3}

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

\frac{x ^{6}}{5} = y \frac{10}{3}

Simplify the expression

\frac{x ^{6}}{5} = \frac{10}{3} y

To test if the graph of \frac{x ^{6}}{5} = \frac{10}{3} y is symmetry with respect to the origin,substitute -x for x and -y for y

\frac{( - x ) ^{6}}{5} = \frac{10}{3} (- y)

Evaluate

\frac{x ^{6}}{5} = \frac{10}{3} (- y)

Multiplying or dividing an odd number of negative terms equals a negative

\frac{x ^{6}}{5} = - \frac{10}{3} y

Solution

Not symmetry with respect to the origin

Show Solution

Rewrite the equation

r = 0 r = \frac{5 50 sin ( θ ) sec ( θ ) \times sec ( θ )}{5 3}

Evaluate

\frac{x ^{6}}{5} = y \times \frac{10}{3}

Use the commutative property to reorder the terms

\frac{x ^{6}}{5} = \frac{10}{3} y

Multiply both sides of the equation by LCD

\frac{x ^{6}}{5} \times 15 = \frac{10}{3} y \times 15

Simplify the equation

More Steps

Evaluate

\frac{x ^{6}}{5} \times 15

Simplify

x^{6} \times 3

Use the commutative property to reorder the terms

3 x^{6}

3 x^{6} = \frac{10}{3} y \times 15

Simplify the equation

More Steps

Evaluate

\frac{10}{3} y \times 15

Simplify

10 y \times 5

Multiply the numbers

50 y

3 x^{6} = 50 y

Move the expression to the left side

3 x^{6} - 50 y = 0

To convert the equation to polar coordinates,substitute x for r cos (θ) and y for r sin (θ)

3 (cos (θ) \times r)^{6} - 50 sin (θ) \times r = 0

Factor the expression

3 cos^{6} (θ) \times r^{6} - 50 sin (θ) \times r = 0

Factor the expression

r (3 cos^{6} (θ) \times r^{5} - 50 sin (θ)) = 0

When the product of factors equals 0,at least one factor is 0

r = 0 3 cos^{6} (θ) \times r^{5} - 50 sin (θ) = 0

Solution

More Steps

Factor the expression

3 cos^{6} (θ) \times r^{5} - 50 sin (θ) = 0

Subtract the terms

3 cos^{6} (θ) \times r^{5} - 50 sin (θ) - (- 50 sin (θ)) = 0 - (- 50 sin (θ))

Evaluate

3 cos^{6} (θ) \times r^{5} = 50 sin (θ)

Divide the terms

r^{5} = \frac{50 sin ( θ )}{3 cos ^{6} ( θ )}

Simplify the expression

r^{5} = \frac{50 sin ( θ ) sec ^{6} ( θ )}{3}

Simplify the expression

More Steps

Evaluate

5 \frac{50 sin ( θ ) sec ^{6} ( θ )}{3}

To take a root of a fraction,take the root of the numerator and denominator separately

\frac{5 50 sin ( θ ) sec ^{6} ( θ )}{5 3}

Simplify the radical expression

\frac{5 50 sin ( θ ) sec ( θ ) \times sec ( θ )}{5 3}

r = \frac{5 50 sin ( θ ) sec ( θ ) \times sec ( θ )}{5 3}

r = 0 r = \frac{5 50 sin ( θ ) sec ( θ ) \times sec ( θ )}{5 3}

Show Solution

Find the first derivative

Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = \frac{9 x ^{5}}{25}

Calculate

\frac{x ^{6}}{5} = y \frac{10}{3}

Simplify the expression

\frac{x ^{6}}{5} = \frac{10}{3} y

Take the derivative of both sides

\frac{d}{d x} (\frac{x ^{6}}{5}) = \frac{d}{d x} (\frac{10}{3} y)

Calculate the derivative

More Steps

Evaluate

\frac{d}{d x} (\frac{x ^{6}}{5})

Rewrite the expression

\frac{\frac{d}{d x} ( x ^{6} )}{5}

Use \frac{d}{d x} x^{n} = n x^{n - 1} to find derivative

\frac{6 x ^{5}}{5}

\frac{6 x ^{5}}{5} = \frac{d}{d x} (\frac{10}{3} y)

Calculate the derivative

More Steps

Evaluate

\frac{d}{d x} (\frac{10}{3} y)

Use differentiation rules

\frac{d}{d y} (\frac{10}{3} y) \times \frac{d y}{d x}

Evaluate the derivative

More Steps

Evaluate

\frac{d}{d y} (\frac{10}{3} y)

Use differentiation rule \frac{d}{d x} (c f (x)) = c \times \frac{d}{d x} (f (x))

\frac{10}{3} \times \frac{d}{d y} (y)

Use \frac{d}{d x} x^{n} = n x^{n - 1} to find derivative

\frac{10}{3} \times 1

Any expression multiplied by 1 remains the same

\frac{10}{3}

\frac{10}{3} \frac{d y}{d x}

\frac{6 x ^{5}}{5} = \frac{10}{3} \frac{d y}{d x}

Swap the sides of the equation

\frac{10}{3} \frac{d y}{d x} = \frac{6 x ^{5}}{5}

Multiply by the reciprocal

\frac{10}{3} \frac{d y}{d x} \times \frac{3}{10} = \frac{6 x ^{5}}{5} \times \frac{3}{10}

Multiply

\frac{d y}{d x} = \frac{6 x ^{5}}{5} \times \frac{3}{10}

Solution

More Steps

Evaluate

\frac{6 x ^{5}}{5} \times \frac{3}{10}

Reduce the numbers

\frac{3 x ^{5}}{5} \times \frac{3}{5}

To multiply the fractions,multiply the numerators and denominators separately

\frac{3 x ^{5} \times 3}{5 \times 5}

Multiply the numbers

\frac{9 x ^{5}}{5 \times 5}

Multiply the numbers

\frac{9 x ^{5}}{25}

\frac{d y}{d x} = \frac{9 x ^{5}}{25}

Show Solution

Find the second derivative

Find the second derivative with respect to x

Find the second derivative with respect to y

\frac{d ^{2} y}{d x ^{2}} = \frac{9 x ^{4}}{5}

Calculate

\frac{x ^{6}}{5} = y \frac{10}{3}

Simplify the expression

\frac{x ^{6}}{5} = \frac{10}{3} y

Take the derivative of both sides

\frac{d}{d x} (\frac{x ^{6}}{5}) = \frac{d}{d x} (\frac{10}{3} y)

Calculate the derivative

More Steps

Evaluate

\frac{d}{d x} (\frac{x ^{6}}{5})

Rewrite the expression

\frac{\frac{d}{d x} ( x ^{6} )}{5}

Use \frac{d}{d x} x^{n} = n x^{n - 1} to find derivative

\frac{6 x ^{5}}{5}

\frac{6 x ^{5}}{5} = \frac{d}{d x} (\frac{10}{3} y)

Calculate the derivative

More Steps

Evaluate

\frac{d}{d x} (\frac{10}{3} y)

Use differentiation rules

\frac{d}{d y} (\frac{10}{3} y) \times \frac{d y}{d x}

Evaluate the derivative

More Steps

Evaluate

\frac{d}{d y} (\frac{10}{3} y)

Use differentiation rule \frac{d}{d x} (c f (x)) = c \times \frac{d}{d x} (f (x))

\frac{10}{3} \times \frac{d}{d y} (y)

Use \frac{d}{d x} x^{n} = n x^{n - 1} to find derivative

\frac{10}{3} \times 1

Any expression multiplied by 1 remains the same

\frac{10}{3}

\frac{10}{3} \frac{d y}{d x}

\frac{6 x ^{5}}{5} = \frac{10}{3} \frac{d y}{d x}

Swap the sides of the equation

\frac{10}{3} \frac{d y}{d x} = \frac{6 x ^{5}}{5}

Multiply by the reciprocal

\frac{10}{3} \frac{d y}{d x} \times \frac{3}{10} = \frac{6 x ^{5}}{5} \times \frac{3}{10}

Multiply

\frac{d y}{d x} = \frac{6 x ^{5}}{5} \times \frac{3}{10}

Multiply

More Steps

Evaluate

\frac{6 x ^{5}}{5} \times \frac{3}{10}

Reduce the numbers

\frac{3 x ^{5}}{5} \times \frac{3}{5}

To multiply the fractions,multiply the numerators and denominators separately

\frac{3 x ^{5} \times 3}{5 \times 5}

Multiply the numbers

\frac{9 x ^{5}}{5 \times 5}

Multiply the numbers

\frac{9 x ^{5}}{25}

\frac{d y}{d x} = \frac{9 x ^{5}}{25}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{9 x ^{5}}{25})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{9 x ^{5}}{25})

Rewrite the expression

\frac{d ^{2} y}{d x ^{2}} = \frac{\frac{d}{d x} ( 9 x ^{5} )}{25}

Evaluate the derivative

More Steps

Evaluate

\frac{d}{d x} (9 x^{5})

Simplify

9 \times \frac{d}{d x} (x^{5})

Rewrite the expression

9 \times 5 x^{4}

Multiply the numbers

45 x^{4}

\frac{d ^{2} y}{d x ^{2}} = \frac{45 x ^{4}}{25}

Solution

\frac{d ^{2} y}{d x ^{2}} = \frac{9 x ^{4}}{5}

Show Solution

Function

Solve the equation

Testing for symmetry

Rewrite the equation

Find the first derivative

Find the second derivative

Graph