Solve (3x)/4 = 5x y -6

Question

Solve the equation

Solve for x

Solve for y

x = - \frac{24}{3 - 20 y}

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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{3 cos ( θ ) - 9 cos ^{2} ( θ ) + 960 sin ( 2 θ )}{20 sin ( 2 θ )} r = \frac{3 cos ( θ ) + 9 cos ^{2} ( θ ) + 960 sin ( 2 θ )}{20 sin ( 2 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{3 - 20 y}{20 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{- 3 + 20 y}{10 x ^{2}}

Calculate

\frac{3 x}{4} = 5 x y - 6

Take the derivative of both sides

\frac{d}{d x} (\frac{3 x}{4}) = \frac{d}{d x} (5 x y - 6)

Calculate the derivative

More Steps

\frac{3}{4} = \frac{d}{d x} (5 x y - 6)

Calculate the derivative

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\frac{3}{4} = 5 y + 5 x \frac{d y}{d x}

Swap the sides of the equation

5 y + 5 x \frac{d y}{d x} = \frac{3}{4}

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

5 x \frac{d y}{d x} = \frac{3}{4} - 5 y

Divide both sides

\frac{5 x \frac{d y}{d x}}{5 x} = \frac{\frac{3}{4} - 5 y}{5 x}

Divide the numbers

\frac{d y}{d x} = \frac{\frac{3}{4} - 5 y}{5 x}

Divide the numbers

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\frac{d y}{d x} = \frac{3 - 20 y}{20 x}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{3 - 20 y}{20 x})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{3 - 20 y}{20 x})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = \frac{\frac{d}{d x} ( 3 - 20 y ) \times 20 x - ( 3 - 20 y ) \times \frac{d}{d x} ( 20 x )}{( 20 x ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{- 20 \frac{d y}{d x} \times 20 x - ( 3 - 20 y ) \times \frac{d}{d x} ( 20 x )}{( 20 x ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{- 20 \frac{d y}{d x} \times 20 x - ( 3 - 20 y ) \times 20}{( 20 x ) ^{2}}

Calculate

\frac{d ^{2} y}{d x ^{2}} = \frac{- 400 \frac{d y}{d x} \times x - ( 3 - 20 y ) \times 20}{( 20 x ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{- 400 \frac{d y}{d x} \times x - ( 60 - 400 y )}{( 20 x ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{- 400 x \frac{d y}{d x} - 60 + 400 y}{( 20 x ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{- 400 x \frac{d y}{d x} - 60 + 400 y}{400 x ^{2}}

Calculate

\frac{d ^{2} y}{d x ^{2}} = \frac{- 20 x \frac{d y}{d x} - 3 + 20 y}{20 x ^{2}}

Use equation \frac{d y}{d x} = \frac{3 - 20 y}{20 x} to substitute

\frac{d ^{2} y}{d x ^{2}} = \frac{- 20 x \times \frac{3 - 20 y}{20 x} - 3 + 20 y}{20 x ^{2}}

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{- 3 + 20 y}{10 x ^{2}}

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