Solve 6x 7y=4x^4y

Question

Solve the equation

Solve for x

Solve for y

x = 0 x = \frac{3 84}{2}

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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

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Rewrite the equation

r = 0 r = \frac{3 21 \times sec ( θ )}{3 2}

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Find the first derivative

Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = \frac{8 y x ^{3} - 21 y}{21 x - 2 x ^{4}}

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Find the second derivative

Find the second derivative with respect to x

Find the second derivative with respect to y

\frac{d ^{2} y}{d x ^{2}} = \frac{882 y - 168 x ^{3} y + 80 x ^{6} y}{441 x ^{2} - 84 x ^{5} + 4 x ^{8}}

Calculate

6 x 7 y = 4 x^{4} y

Simplify the expression

42 x y = 4 x^{4} y

Take the derivative of both sides

\frac{d}{d x} (42 x y) = \frac{d}{d x} (4 x^{4} y)

Calculate the derivative

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42 y + 42 x \frac{d y}{d x} = \frac{d}{d x} (4 x^{4} y)

Calculate the derivative

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42 y + 42 x \frac{d y}{d x} = 16 x^{3} y + 4 x^{4} \frac{d y}{d x}

Move the expression to the left side

42 y + 42 x \frac{d y}{d x} - 4 x^{4} \frac{d y}{d x} = 16 x^{3} y

Move the expression to the right side

42 x \frac{d y}{d x} - 4 x^{4} \frac{d y}{d x} = 16 x^{3} y - 42 y

Collect like terms by calculating the sum or difference of their coefficients

(42 x - 4 x^{4}) \frac{d y}{d x} = 16 x^{3} y - 42 y

Divide both sides

\frac{( 42 x - 4 x ^{4} ) \frac{d y}{d x}}{42 x - 4 x ^{4}} = \frac{16 x ^{3} y - 42 y}{42 x - 4 x ^{4}}

Divide the numbers

\frac{d y}{d x} = \frac{16 x ^{3} y - 42 y}{42 x - 4 x ^{4}}

Divide the numbers

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\frac{d y}{d x} = \frac{8 y x ^{3} - 21 y}{21 x - 2 x ^{4}}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{8 y x ^{3} - 21 y}{21 x - 2 x ^{4}})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{8 y x ^{3} - 21 y}{21 x - 2 x ^{4}})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = \frac{\frac{d}{d x} ( 8 y x ^{3} - 21 y ) \times ( 21 x - 2 x ^{4} ) - ( 8 y x ^{3} - 21 y ) \times \frac{d}{d x} ( 21 x - 2 x ^{4} )}{( 21 x - 2 x ^{4} ) ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = \frac{( 24 x ^{2} y + 8 x ^{3} \frac{d y}{d x} - 21 \frac{d y}{d x} ) ( 21 x - 2 x ^{4} ) - ( 8 y x ^{3} - 21 y ) \times \frac{d}{d x} ( 21 x - 2 x ^{4} )}{( 21 x - 2 x ^{4} ) ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = \frac{( 24 x ^{2} y + 8 x ^{3} \frac{d y}{d x} - 21 \frac{d y}{d x} ) ( 21 x - 2 x ^{4} ) - ( 8 y x ^{3} - 21 y ) ( 21 - 8 x ^{3} )}{( 21 x - 2 x ^{4} ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{504 x ^{3} y - 48 x ^{6} y + 210 x ^{4} \frac{d y}{d x} - 16 x ^{7} \frac{d y}{d x} - 441 x \frac{d y}{d x} - ( 8 y x ^{3} - 21 y ) ( 21 - 8 x ^{3} )}{( 21 x - 2 x ^{4} ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{504 x ^{3} y - 48 x ^{6} y + 210 x ^{4} \frac{d y}{d x} - 16 x ^{7} \frac{d y}{d x} - 441 x \frac{d y}{d x} - ( 336 y x ^{3} - 441 y - 64 y x ^{6} )}{( 21 x - 2 x ^{4} ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{168 x ^{3} y + 16 x ^{6} y + 210 x ^{4} \frac{d y}{d x} - 16 x ^{7} \frac{d y}{d x} - 441 x \frac{d y}{d x} + 441 y}{( 21 x - 2 x ^{4} ) ^{2}}

Use equation \frac{d y}{d x} = \frac{8 y x ^{3} - 21 y}{21 x - 2 x ^{4}} to substitute

\frac{d ^{2} y}{d x ^{2}} = \frac{168 x ^{3} y + 16 x ^{6} y + 210 x ^{4} \times \frac{8 y x ^{3} - 21 y}{21 x - 2 x ^{4}} - 16 x ^{7} \times \frac{8 y x ^{3} - 21 y}{21 x - 2 x ^{4}} - 441 x \times \frac{8 y x ^{3} - 21 y}{21 x - 2 x ^{4}} + 441 y}{( 21 x - 2 x ^{4} ) ^{2}}

Solution

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Calculate

\frac{168 x ^{3} y + 16 x ^{6} y + 210 x ^{4} \times \frac{8 y x ^{3} - 21 y}{21 x - 2 x ^{4}} - 16 x ^{7} \times \frac{8 y x ^{3} - 21 y}{21 x - 2 x ^{4}} - 441 x \times \frac{8 y x ^{3} - 21 y}{21 x - 2 x ^{4}} + 441 y}{( 21 x - 2 x ^{4} ) ^{2}}

Multiply the terms

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\frac{168 x ^{3} y + 16 x ^{6} y + \frac{210 x ^{3} ( 8 y x ^{3} - 21 y )}{21 - 2 x ^{3}} - 16 x ^{7} \times \frac{8 y x ^{3} - 21 y}{21 x - 2 x ^{4}} - 441 x \times \frac{8 y x ^{3} - 21 y}{21 x - 2 x ^{4}} + 441 y}{( 21 x - 2 x ^{4} ) ^{2}}

Multiply the terms

\frac{168 x ^{3} y + 16 x ^{6} y + \frac{210 x ^{3} ( 8 y x ^{3} - 21 y )}{21 - 2 x ^{3}} - \frac{16 x ^{6} ( 8 y x ^{3} - 21 y )}{21 - 2 x ^{3}} - 441 x \times \frac{8 y x ^{3} - 21 y}{21 x - 2 x ^{4}} + 441 y}{( 21 x - 2 x ^{4} ) ^{2}}

Multiply the terms

\frac{168 x ^{3} y + 16 x ^{6} y + \frac{210 x ^{3} ( 8 y x ^{3} - 21 y )}{21 - 2 x ^{3}} - \frac{16 x ^{6} ( 8 y x ^{3} - 21 y )}{21 - 2 x ^{3}} - \frac{441 ( 8 y x ^{3} - 21 y )}{21 - 2 x ^{3}} + 441 y}{( 21 x - 2 x ^{4} ) ^{2}}

Calculate the sum or difference

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\frac{882 y - 168 x ^{3} y + 80 x ^{6} y}{( 21 x - 2 x ^{4} ) ^{2}}

Expand the expression

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\frac{882 y - 168 x ^{3} y + 80 x ^{6} y}{441 x ^{2} - 84 x ^{5} + 4 x ^{8}}

\frac{d ^{2} y}{d x ^{2}} = \frac{882 y - 168 x ^{3} y + 80 x ^{6} y}{441 x ^{2} - 84 x ^{5} + 4 x ^{8}}

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