Solve x 7x^21/2y^4=3

Question

Solve the equation

x = \frac{3 294 y ^{2}}{7 y ^{2}}

Show Solution

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

Show Solution

r = \frac{7 6}{7 7 cos ^{3} ( θ ) sin ^{4} ( θ )}

Show Solution

Find the derivative with respect to x

Find the derivative with respect to y

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

Show Solution

Find the second derivative with respect to x

Find the second derivative with respect to y

\frac{d ^{2} y}{d x ^{2}} = \frac{21 y}{16 x ^{2}}

Calculate

x 7 x^{2} \frac{1}{2} y^{4} = 3

Simplify the expression

\frac{7}{2} x^{3} y^{4} = 3

Take the derivative of both sides

\frac{d}{d x} (\frac{7}{2} x^{3} y^{4}) = \frac{d}{d x} (3)

Calculate the derivative

More Steps

\frac{21}{2} x^{2} y^{4} + 14 x^{3} y^{3} \frac{d y}{d x} = \frac{d}{d x} (3)

Calculate the derivative

\frac{21}{2} x^{2} y^{4} + 14 x^{3} y^{3} \frac{d y}{d x} = 0

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

14 x^{3} y^{3} \frac{d y}{d x} = 0 - \frac{21}{2} x^{2} y^{4}

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

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

Divide both sides

\frac{14 x ^{3} y ^{3} \frac{d y}{d x}}{14 x ^{3} y ^{3}} = \frac{- \frac{21}{2} x ^{2} y ^{4}}{14 x ^{3} y ^{3}}

Divide the numbers

\frac{d y}{d x} = \frac{- \frac{21}{2} x ^{2} y ^{4}}{14 x ^{3} y ^{3}}

Divide the numbers

More Steps

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

More Steps

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

Calculate the derivative

More Steps

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

Calculate

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

Calculate

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

Use the commutative property to reorder the terms

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

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{12 x \frac{d y}{d x} - 12 y}{16 x ^{2}}

Calculate

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

Use equation \frac{d y}{d x} = - \frac{3 y}{4 x} to substitute

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

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{21 y}{16 x ^{2}}

Show Solution