Solve sin(xy)=x y

Question

Testing for symmetry

Testing for symmetry about the origin

Testing for symmetry about the x-axis

Testing for symmetry about the y-axis

Symmetry with respect to the origin

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Find the derivative with respect to x

Find the derivative with respect to y

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

Calculate

sin (x y) = x y

Take the derivative of both sides

\frac{d}{d x} (sin (x y)) = \frac{d}{d x} (x y)

Calculate the derivative

(y + x \frac{d y}{d x}) cos (x y) = \frac{d}{d x} (x y)

Calculate the derivative

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(y + x \frac{d y}{d x}) cos (x y) = y + x \frac{d y}{d x}

Rewrite the expression

cos (x y) (y + x \frac{d y}{d x}) = y + x \frac{d y}{d x}

Expand the expression

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y cos (x y) + x cos (x y) \frac{d y}{d x} = y + x \frac{d y}{d x}

Move the expression to the left side

y cos (x y) + x cos (x y) \frac{d y}{d x} - x \frac{d y}{d x} = y

Move the expression to the right side

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

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

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

Divide both sides

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

Divide the numbers

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

Divide the numbers

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

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

Use the commutative property to reorder the terms

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

Any expression multiplied by 1 remains the same

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

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

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

Solution

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

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