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e3practice

# e3practice - = | 2(d/dx(x(x^2-y^2 | Use the quotient rule...

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1)(10) Does f ( x, y ) = ln( x 2 −y 2 ) satisfy the partial differential equation below? ∂f/∂x^2 - ∂f/∂y^2 = 0. Possible derivation: d/dx(log(x^2-y^2)) | Use the chain rule, d/dx(log(x^2-y^2)) = ( dlog(u))/( du) ( du)/( dx), where u = x^2-y^2 and ( dlog(u))/( du) = 1/u: = | (d/dx(x^2-y^2))/(x^2-y^2) | Differentiate the sum term by term and factor out constants: = | (d/dx(x^2)-d/dx(y^2))/(x^2-y^2) | The derivative of x^2 is 2 x: = | (2 x-d/dx(y^2))/(x^2-y^2) | The derivative of y^2 is zero: = | (2 x-0)/(x^2-y^2) Possible derivation: d/dx((2 x)/(x^2-y^2)) | Factor out constants:
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Unformatted text preview: = | 2 (d/dx(x/(x^2-y^2))) | Use the quotient rule, d/dx(u/v) = (v ( du)/( dx)-u ( dv)/( dx))/v^2, where u = x and v = x^2-y^2: = | 2 ((x^2-y^2) (d/dx(x))-x (d/dx(x^2-y^2)))/(x^2-y^2)^2 | The derivative of x is 1: = | (2 (1 (x^2-y^2)-x (d/dx(x^2-y^2))))/(x^2-y^2)^2 | Differentiate the sum term by term and factor out constants: = | (2 (-x (d/dx(x^2)-d/dx(y^2))+x^2-y^2))/(x^2-y^2)^2 | The derivative of x^2 is 2 x: = | (2 (-x (2 x-d/dx(y^2))+x^2-y^2))/(x^2-y^2)^2 | The derivative of y^2 is zero: = | (2 (x^2-(2 x-0) x-y^2))/(x^2-y^2)^2...
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