f(x,y) = In (x sin y) + sin (x In y) find second order derivatives

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To find the second-order derivatives of the function f(x,y)=ln(xsiny)+sin(xlny), we need to take the partial derivatives twice with respect to each variable.Let&#039;s start by finding the first-order partial derivatives:∂f∂x=1xsiny·∂∂x(xsiny)+cos(xlny)·∂∂x(xlny)Simplifying:∂f∂x=sinyxsiny+cos(xlny)·(lny+x·1y)∂f∂x=1x+cos(xlny)·lny+xcos(xlny)yNow, let&#039;s find the second-order partial derivative with respect to x:∂2f∂x2=∂∂x(1x+cos(xlny)·lny+xcos(xlny)y)Differentiating each term separately:∂2f∂x2=−1x2−sin(xlny)·lny+cos(xlny)·ln2y+cos(xlny)·1ySimilarly, let&#039;s find the first-order partial derivative with respect to y:∂f∂y=1xsiny·∂∂y(xsiny)+cos(xlny)·∂∂y(xlny)Simplifying:∂f∂y=xcosyxsiny+cos(xlny)·(1y·x)∂f∂y=cosysiny+xcos(xlny)yFinally, let&#039;s find the second-order partial derivative with respect to y:∂2f∂y2=∂∂y(cosysiny+xcos(xlny)y)Differentiating each term separately:∂2f∂y2=−cosysin2y−xcos(xlny)y2−x2sin(xlny)y2

f(x,y) = In (x sin y) + sin

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