Solve 12y^6x=72 - ScanMath

Question

Solve the equation

Solve for x

Solve for y

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

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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 = 7 6 csc^{6} (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}{6 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{7 y}{36 x ^{2}}

Calculate

12 y^{6} x = 72

Take the derivative of both sides

\frac{d}{d x} (12 y^{6} x) = \frac{d}{d x} (72)

Calculate the derivative

More Steps

12 y^{6} + 72 x y^{5} \frac{d y}{d x} = \frac{d}{d x} (72)

Calculate the derivative

12 y^{6} + 72 x y^{5} \frac{d y}{d x} = 0

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

72 x y^{5} \frac{d y}{d x} = 0 - 12 y^{6}

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

72 x y^{5} \frac{d y}{d x} = - 12 y^{6}

Divide both sides

\frac{72 x y ^{5} \frac{d y}{d x}}{72 x y ^{5}} = \frac{- 12 y ^{6}}{72 x y ^{5}}

Divide the numbers

\frac{d y}{d x} = \frac{- 12 y ^{6}}{72 x y ^{5}}

Divide the numbers

More Steps

\frac{d y}{d x} = - \frac{y}{6 x}

Take the derivative of both sides

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

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (- \frac{y}{6 x})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = - \frac{\frac{d}{d x} ( y ) \times 6 x - y \times \frac{d}{d x} ( 6 x )}{( 6 x ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{\frac{d y}{d x} \times 6 x - y \times \frac{d}{d x} ( 6 x )}{( 6 x ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{\frac{d y}{d x} \times 6 x - y \times 6}{( 6 x ) ^{2}}

Use the commutative property to reorder the terms

\frac{d ^{2} y}{d x ^{2}} = - \frac{6 \frac{d y}{d x} \times x - y \times 6}{( 6 x ) ^{2}}

Use the commutative property to reorder the terms

\frac{d ^{2} y}{d x ^{2}} = - \frac{6 \frac{d y}{d x} \times x - 6 y}{( 6 x ) ^{2}}

Use the commutative property to reorder the terms

\frac{d ^{2} y}{d x ^{2}} = - \frac{6 x \frac{d y}{d x} - 6 y}{( 6 x ) ^{2}}

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{6 x \frac{d y}{d x} - 6 y}{36 x ^{2}}

Calculate

\frac{d ^{2} y}{d x ^{2}} = - \frac{x \frac{d y}{d x} - y}{6 x ^{2}}

Use equation \frac{d y}{d x} = - \frac{y}{6 x} to substitute

\frac{d ^{2} y}{d x ^{2}} = - \frac{x ( - \frac{y}{6 x} ) - y}{6 x ^{2}}

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{7 y}{36 x ^{2}}

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