Solve 4x^6y-5=12 - ScanMath

Solve the equation

y = \frac{17}{4 x ^{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 = \frac{7 17}{7 4 cos ^{6} ( θ ) sin ( 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{6 y}{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{42 y}{x ^{2}}

Calculate

4 x^{6} y - 5 = 12

Take the derivative of both sides

\frac{d}{d x} (4 x^{6} y - 5) = \frac{d}{d x} (12)

Calculate the derivative

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24 x^{5} y + 4 x^{6} \frac{d y}{d x} = \frac{d}{d x} (12)

Calculate the derivative

24 x^{5} y + 4 x^{6} \frac{d y}{d x} = 0

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

4 x^{6} \frac{d y}{d x} = 0 - 24 x^{5} y

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

4 x^{6} \frac{d y}{d x} = - 24 x^{5} y

Divide both sides

\frac{4 x ^{6} \frac{d y}{d x}}{4 x ^{6}} = \frac{- 24 x ^{5} y}{4 x ^{6}}

Divide the numbers

\frac{d y}{d x} = \frac{- 24 x ^{5} y}{4 x ^{6}}

Divide the numbers

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

More Steps

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

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

\frac{d ^{2} y}{d x ^{2}} = - \frac{6 \frac{d y}{d x} \times x - 6 y \times 1}{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 \times 1}{x ^{2}}

Any expression multiplied by 1 remains the same

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

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

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

Solution

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\frac{d ^{2} y}{d x ^{2}} = \frac{42 y}{x ^{2}}

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