Solve (x-3)^2 (y^4)^2 = 25

Question

Solve the equation

Solve for x

Solve for y

x = \frac{5 + 3 y ^{4}}{y ^{4}} x = \frac{- 5 + 3 y ^{4}}{y ^{4}}

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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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Find the first derivative

Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = - \frac{y}{4 x - 12}

Calculate

(x - 3)^{2} (y^{4})^{2} = 25

Simplify the expression

y^{8} (x - 3)^{2} = 25

Take the derivative of both sides

\frac{d}{d x} (y^{8} (x - 3)^{2}) = \frac{d}{d x} (25)

Calculate the derivative

More Steps

8 y^{7} \frac{d y}{d x} \times (x - 3)^{2} + 2 y^{8} x - 6 y^{8} = \frac{d}{d x} (25)

Calculate the derivative

8 y^{7} \frac{d y}{d x} \times (x - 3)^{2} + 2 y^{8} x - 6 y^{8} = 0

Rewrite the expression

8 (x - 3)^{2} y^{7} \frac{d y}{d x} + 2 y^{8} x - 6 y^{8} = 0

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

8 (x - 3)^{2} y^{7} \frac{d y}{d x} = 0 - (2 y^{8} x - 6 y^{8})

Subtract the terms

More Steps

8 (x - 3)^{2} y^{7} \frac{d y}{d x} = - 2 y^{8} x + 6 y^{8}

Divide both sides

\frac{8 ( x - 3 ) ^{2} y ^{7} \frac{d y}{d x}}{8 ( x - 3 ) ^{2} y ^{7}} = \frac{- 2 y ^{8} x + 6 y ^{8}}{8 ( x - 3 ) ^{2} y ^{7}}

Divide the numbers

\frac{d y}{d x} = \frac{- 2 y ^{8} x + 6 y ^{8}}{8 ( x - 3 ) ^{2} y ^{7}}

Divide the numbers

More Steps

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

Expand the expression

More Steps

\frac{d y}{d x} = \frac{- x y + 3 y}{4 x ^{2} - 24 x + 36}

Solution

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\frac{d y}{d x} = - \frac{y}{4 x - 12}

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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{5 y}{16 x ^{2} - 96 x + 144}

Calculate

(x - 3)^{2} (y^{4})^{2} = 25

Simplify the expression

y^{8} (x - 3)^{2} = 25

Take the derivative of both sides

\frac{d}{d x} (y^{8} (x - 3)^{2}) = \frac{d}{d x} (25)

Calculate the derivative

More Steps

8 y^{7} \frac{d y}{d x} \times (x - 3)^{2} + 2 y^{8} x - 6 y^{8} = \frac{d}{d x} (25)

Calculate the derivative

8 y^{7} \frac{d y}{d x} \times (x - 3)^{2} + 2 y^{8} x - 6 y^{8} = 0

Rewrite the expression

8 (x - 3)^{2} y^{7} \frac{d y}{d x} + 2 y^{8} x - 6 y^{8} = 0

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

8 (x - 3)^{2} y^{7} \frac{d y}{d x} = 0 - (2 y^{8} x - 6 y^{8})

Subtract the terms

More Steps

8 (x - 3)^{2} y^{7} \frac{d y}{d x} = - 2 y^{8} x + 6 y^{8}

Divide both sides

\frac{8 ( x - 3 ) ^{2} y ^{7} \frac{d y}{d x}}{8 ( x - 3 ) ^{2} y ^{7}} = \frac{- 2 y ^{8} x + 6 y ^{8}}{8 ( x - 3 ) ^{2} y ^{7}}

Divide the numbers

\frac{d y}{d x} = \frac{- 2 y ^{8} x + 6 y ^{8}}{8 ( x - 3 ) ^{2} y ^{7}}

Divide the numbers

More Steps

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

Expand the expression

More Steps

\frac{d y}{d x} = \frac{- x y + 3 y}{4 x ^{2} - 24 x + 36}

Simplify

More Steps

\frac{d y}{d x} = - \frac{y}{4 x - 12}

Take the derivative of both sides

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

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (- \frac{y}{4 x - 12})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = - \frac{\frac{d}{d x} ( y ) \times ( 4 x - 12 ) - y \times \frac{d}{d x} ( 4 x - 12 )}{( 4 x - 12 ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{\frac{d y}{d x} \times ( 4 x - 12 ) - y \times \frac{d}{d x} ( 4 x - 12 )}{( 4 x - 12 ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{\frac{d y}{d x} \times ( 4 x - 12 ) - y \times 4}{( 4 x - 12 ) ^{2}}

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{4 x \frac{d y}{d x} - 12 \frac{d y}{d x} - y \times 4}{( 4 x - 12 ) ^{2}}

Use the commutative property to reorder the terms

\frac{d ^{2} y}{d x ^{2}} = - \frac{4 x \frac{d y}{d x} - 12 \frac{d y}{d x} - 4 y}{( 4 x - 12 ) ^{2}}

Calculate

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

Use equation \frac{d y}{d x} = - \frac{y}{4 x - 12} to substitute

\frac{d ^{2} y}{d x ^{2}} = - \frac{x ( - \frac{y}{4 x - 12} ) - 3 ( - \frac{y}{4 x - 12} ) - y}{4 ( x - 3 ) ^{2}}

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{5 y}{16 x ^{2} - 96 x + 144}

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