Solve y^4x-8=0 - ScanMath

Question

Solve the equation

Solve for x

Solve for y

x = \frac{8}{y ^{4}}

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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

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r = 5 8 csc^{4} (θ) sec (θ)

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Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = - \frac{y}{4 x}

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Find the second derivative with respect to x

Find the second derivative with respect to y

\frac{d ^{2} y}{d x ^{2}} = \frac{5 y}{16 x ^{2}}

Calculate

y^{4} x - 8 = 0

Take the derivative of both sides

\frac{d}{d x} (y^{4} x - 8) = \frac{d}{d x} (0)

Calculate the derivative

More Steps

y^{4} + 4 x y^{3} \frac{d y}{d x} = \frac{d}{d x} (0)

Calculate the derivative

y^{4} + 4 x y^{3} \frac{d y}{d x} = 0

Move the expression to the right-hand side and change its sign

4 x y^{3} \frac{d y}{d x} = 0 - y^{4}

Removing 0 doesn't change the value,so remove it from the expression

4 x y^{3} \frac{d y}{d x} = - y^{4}

Divide both sides

\frac{4 x y ^{3} \frac{d y}{d x}}{4 x y ^{3}} = \frac{- y ^{4}}{4 x y ^{3}}

Divide the numbers

\frac{d y}{d x} = \frac{- y ^{4}}{4 x y ^{3}}

Divide the numbers

More Steps

\frac{d y}{d x} = - \frac{y}{4 x}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (- \frac{y}{4 x})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (- \frac{y}{4 x})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = - \frac{\frac{d}{d x} ( y ) \times 4 x - y \times \frac{d}{d x} ( 4 x )}{( 4 x ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{\frac{d y}{d x} \times 4 x - y \times \frac{d}{d x} ( 4 x )}{( 4 x ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{\frac{d y}{d x} \times 4 x - y \times 4}{( 4 x ) ^{2}}

Use the commutative property to reorder the terms

\frac{d ^{2} y}{d x ^{2}} = - \frac{4 \frac{d y}{d x} \times x - y \times 4}{( 4 x ) ^{2}}

Use the commutative property to reorder the terms

\frac{d ^{2} y}{d x ^{2}} = - \frac{4 \frac{d y}{d x} \times x - 4 y}{( 4 x ) ^{2}}

Use the commutative property to reorder the terms

\frac{d ^{2} y}{d x ^{2}} = - \frac{4 x \frac{d y}{d x} - 4 y}{( 4 x ) ^{2}}

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{4 x \frac{d y}{d x} - 4 y}{16 x ^{2}}

Calculate

\frac{d ^{2} y}{d x ^{2}} = - \frac{x \frac{d y}{d x} - y}{4 x ^{2}}

Use equation \frac{d y}{d x} = - \frac{y}{4 x} to substitute

\frac{d ^{2} y}{d x ^{2}} = - \frac{x ( - \frac{y}{4 x} ) - y}{4 x ^{2}}

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{5 y}{16 x ^{2}}

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