Solve sin(x y)=x - ScanMath

Question

Solve the equation

y = \frac{arcsin ( x )}{x}

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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{1 - y cos ( x y )}{x cos ( x y )}

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Find the second derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{( - 1 + y cos ( x y ) ) cos ^{2} ( x y ) + x sin ( x y ) + cos ^{3} ( x y ) \times y - cos ^{2} ( x y )}{cos ^{3} ( x y ) \times x ^{2}}

Calculate

sin (x y) = x

Take the derivative of both sides

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

Calculate the derivative

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

Use \frac{d}{d x} x^{n} = n x^{n - 1} to find derivative

(y + x \frac{d y}{d x}) cos (x y) = 1

Rewrite the expression

cos (x y) (y + x \frac{d y}{d x}) = 1

Divide both sides

\frac{cos ( x y ) ( y + x \frac{d y}{d x} )}{cos ( x y )} = \frac{1}{cos ( x y )}

Divide the numbers

y + x \frac{d y}{d x} = \frac{1}{cos ( x y )}

Move the constant to the right side

x \frac{d y}{d x} = \frac{1}{cos ( x y )} - y

Subtract the terms

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x \frac{d y}{d x} = \frac{1 - y cos ( x y )}{cos ( x y )}

Multiply by the reciprocal

x \frac{d y}{d x} \times \frac{1}{x} = \frac{1 - y cos ( x y )}{cos ( x y )} \times \frac{1}{x}

Multiply

\frac{d y}{d x} = \frac{1 - y cos ( x y )}{cos ( x y )} \times \frac{1}{x}

Multiply

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\frac{d y}{d x} = \frac{1 - y cos ( x y )}{x cos ( x y )}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{1 - y cos ( x y )}{x cos ( x y )})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{1 - y cos ( x y )}{x cos ( x y )})

Use differentiation rules

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

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = \frac{( - \frac{d y}{d x} \times cos ( x y ) - y ( - y - x \frac{d y}{d x} ) sin ( x y ) ) x cos ( x y ) - ( 1 - y cos ( x y ) ) \times \frac{d}{d x} ( x cos ( x y ) )}{( x cos ( x y ) ) ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = \frac{( - \frac{d y}{d x} \times cos ( x y ) - y ( - y - x \frac{d y}{d x} ) sin ( x y ) ) x cos ( x y ) - ( 1 - y cos ( x y ) ) ( cos ( x y ) + x ( - y - x \frac{d y}{d x} ) sin ( x y ) )}{( x cos ( x y ) ) ^{2}}

Calculate

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

Calculate

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

Calculate

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

Use equation \frac{d y}{d x} = \frac{1 - y cos ( x y )}{x cos ( x y )} to substitute

\frac{d ^{2} y}{d x ^{2}} = \frac{- x \times \frac{1 - y c o s ( x y )}{x c o s ( x y )} \times cos ^{2} ( x y ) - cos ( x y ) - x ( - y - x \times \frac{1 - y c o s ( x y )}{x c o s ( x y )} ) sin ( x y ) + y cos ^{2} ( x y )}{cos ^{2} ( x y ) x ^{2}}

Solution

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Calculate

\frac{- x \times \frac{1 - y c o s ( x y )}{x c o s ( x y )} \times cos ^{2} ( x y ) - cos ( x y ) - x ( - y - x \times \frac{1 - y c o s ( x y )}{x c o s ( x y )} ) sin ( x y ) + y cos ^{2} ( x y )}{cos ^{2} ( x y ) x ^{2}}

Evaluate

\frac{- x \times \frac{1 - y c o s ( x y )}{x c o s ( x y )} \times cos ^{2} ( x y ) - cos ( x y ) - x ( - y - x \times \frac{1 - y c o s ( x y )}{x c o s ( x y )} ) sin ( x y ) + y cos ^{2} ( x y )}{cos ^{2} ( x y ) \times x ^{2}}

Multiply the terms

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\frac{- x \times \frac{1 - y c o s ( x y )}{x c o s ( x y )} \times cos ^{2} ( x y ) - cos ( x y ) - x ( - y - \frac{1 - y c o s ( x y )}{c o s ( x y )} ) sin ( x y ) + y cos ^{2} ( x y )}{cos ^{2} ( x y ) \times x ^{2}}

Subtract the terms

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\frac{- x \times \frac{1 - y c o s ( x y )}{x c o s ( x y )} \times cos ^{2} ( x y ) - cos ( x y ) - x ( - \frac{1}{c o s ( x y )} ) sin ( x y ) + y cos ^{2} ( x y )}{cos ^{2} ( x y ) \times x ^{2}}

Multiply the terms

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\frac{( - 1 + y cos ( x y ) ) cos ( x y ) - cos ( x y ) - x ( - \frac{1}{c o s ( x y )} ) sin ( x y ) + y cos ^{2} ( x y )}{cos ^{2} ( x y ) \times x ^{2}}

Multiply

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\frac{( - 1 + y cos ( x y ) ) cos ( x y ) - cos ( x y ) + \frac{x s i n ( x y )}{c o s ( x y )} + y cos ^{2} ( x y )}{cos ^{2} ( x y ) \times x ^{2}}

Multiply the terms

\frac{( - 1 + y cos ( x y ) ) cos ( x y ) - cos ( x y ) + \frac{x s i n ( x y )}{c o s ( x y )} + y cos ^{2} ( x y )}{( x cos ( x y ) ) ^{2}}

Calculate the sum or difference

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\frac{\frac{( - 1 + y c o s ( x y ) ) c o s ^{2} ( x y ) + x s i n ( x y ) + y c o s ^{3} ( x y ) - c o s ^{2} ( x y )}{c o s ( x y )}}{( x cos ( x y ) ) ^{2}}

Multiply by the reciprocal

\frac{( - 1 + y cos ( x y ) ) cos ^{2} ( x y ) + x sin ( x y ) + y cos ^{3} ( x y ) - cos ^{2} ( x y )}{cos ( x y )} \times \frac{1}{( x cos ( x y ) ) ^{2}}

Multiply the terms

\frac{( - 1 + y cos ( x y ) ) cos ^{2} ( x y ) + x sin ( x y ) + y cos ^{3} ( x y ) - cos ^{2} ( x y )}{cos ( x y ) ( x cos ( x y ) ) ^{2}}

Calculate

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\frac{( - 1 + y cos ( x y ) ) cos ^{2} ( x y ) + x sin ( x y ) + y cos ^{3} ( x y ) - cos ^{2} ( x y )}{cos ^{3} ( x y ) \times x ^{2}}

Multiply the terms

\frac{( - 1 + y cos ( x y ) ) cos ^{2} ( x y ) + x sin ( x y ) + cos ^{3} ( x y ) \times y - cos ^{2} ( x y )}{cos ^{3} ( x y ) \times x ^{2}}

\frac{d ^{2} y}{d x ^{2}} = \frac{( - 1 + y cos ( x y ) ) cos ^{2} ( x y ) + x sin ( x y ) + cos ^{3} ( x y ) \times y - cos ^{2} ( x y )}{cos ^{3} ( x y ) \times x ^{2}}

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