Solve y =- 2x^220x^42

Question

Function

Evaluate the derivative

Find the domain

Find the x -intercept/zero

y^{'} = - 480 x^{5}

Evaluate

y = - 2 x^{2} \times 20 x^{4} \times 2

Simplify

More Steps

Evaluate

- 2 x^{2} \times 20 x^{4} \times 2

Multiply the terms

More Steps

Evaluate

2 \times 20 \times 2

Multiply the terms

40 \times 2

Multiply the numbers

80

- 80 x^{2} \times x^{4}

Multiply the terms with the same base by adding their exponents

- 80 x^{2 + 4}

Add the numbers

- 80 x^{6}

y = - 80 x^{6}

Take the derivative of both sides

y^{'} = \frac{d}{d x} (- 80 x^{6})

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

y^{'} = - 80 \times \frac{d}{d x} (x^{6})

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

y^{'} = - 80 \times 6 x^{5}

Solution

y^{'} = - 480 x^{5}

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

y = - 2 x^{2} 20 x^{4} 2

Simplify the expression

y = - 80 x^{6}

To test if the graph of y = - 80 x^{6} is symmetry with respect to the origin,substitute -x for x and -y for y

- y = - 80 (- x)^{6}

Simplify

- y = - 80 x^{6}

Change the signs both sides

y = 80 x^{6}

Solution

Not symmetry with respect to the origin

Show Solution

Solve the equation

Solve for x

Solve for y

x = \frac{6 - 8 0 ^{5} y}{80} x = - \frac{6 - 8 0 ^{5} y}{80}

Evaluate

y = - 2 x^{2} \times 20 x^{4} \times 2

Simplify

More Steps

Evaluate

- 2 x^{2} \times 20 x^{4} \times 2

Multiply the terms

More Steps

Evaluate

2 \times 20 \times 2

Multiply the terms

40 \times 2

Multiply the numbers

80

- 80 x^{2} \times x^{4}

Multiply the terms with the same base by adding their exponents

- 80 x^{2 + 4}

Add the numbers

- 80 x^{6}

y = - 80 x^{6}

Swap the sides of the equation

- 80 x^{6} = y

Change the signs on both sides of the equation

80 x^{6} = - y

Divide both sides

\frac{80 x ^{6}}{80} = \frac{- y}{80}

Divide the numbers

x^{6} = \frac{- y}{80}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

x^{6} = - \frac{y}{80}

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

x = \pm 6 - \frac{y}{80}

Simplify the expression

More Steps

Evaluate

6 - \frac{y}{80}

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

\frac{6 - y}{6 80}

Multiply by the Conjugate

\frac{6 - y \times 6 8 0 ^{5}}{6 80 \times 6 8 0 ^{5}}

Calculate

\frac{6 - y \times 6 8 0 ^{5}}{80}

Calculate

More Steps

Evaluate

6 - y \times 6 8 0^{5}

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

6 - y \times 8 0^{5}

Calculate the product

6 - 8 0^{5} y

\frac{6 - 8 0 ^{5} y}{80}

x = \pm \frac{6 - 8 0 ^{5} y}{80}

Solution

x = \frac{6 - 8 0 ^{5} y}{80} x = - \frac{6 - 8 0 ^{5} y}{80}

Show Solution

Rewrite the equation

r = 0 r = - \frac{5 sin ( θ )}{5 80 cos ( θ ) \times cos ( θ )}

Evaluate

y = - 2 x^{2} \times 20 x^{4} \times 2

Simplify

More Steps

Evaluate

- 2 x^{2} \times 20 x^{4} \times 2

Multiply the terms

More Steps

Evaluate

2 \times 20 \times 2

Multiply the terms

40 \times 2

Multiply the numbers

80

- 80 x^{2} \times x^{4}

Multiply the terms with the same base by adding their exponents

- 80 x^{2 + 4}

Add the numbers

- 80 x^{6}

y = - 80 x^{6}

Move the expression to the left side

y + 80 x^{6} = 0

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

sin (θ) \times r + 80 (cos (θ) \times r)^{6} = 0

Factor the expression

80 cos^{6} (θ) \times r^{6} + sin (θ) \times r = 0

Factor the expression

r (80 cos^{6} (θ) \times r^{5} + sin (θ)) = 0

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

r = 0 80 cos^{6} (θ) \times r^{5} + sin (θ) = 0

Solution

More Steps

Factor the expression

80 cos^{6} (θ) \times r^{5} + sin (θ) = 0

Subtract the terms

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

Evaluate

80 cos^{6} (θ) \times r^{5} = - sin (θ)

Divide the terms

r^{5} = - \frac{sin ( θ )}{80 cos ^{6} ( θ )}

Simplify the expression

More Steps

Evaluate

5 - \frac{sin ( θ )}{80 cos ^{6} ( θ )}

An odd root of a negative radicand is always a negative

- 5 \frac{sin ( θ )}{80 cos ^{6} ( θ )}

Simplify the radical expression

- \frac{5 sin ( θ )}{5 80 cos ( θ ) \times cos ( θ )}

r = - \frac{5 sin ( θ )}{5 80 cos ( θ ) \times cos ( θ )}

r = 0 r = - \frac{5 sin ( θ )}{5 80 cos ( θ ) \times cos ( θ )}

Show Solution

Function

Testing for symmetry

Solve the equation

Rewrite the equation

Graph