Answer to Question #135766 in Calculus for Promise Omiponle

Question #135766
5. Find all the first and second partial derivatives of f(x,y)=x^(2) y + cos(xy) + tan(x/y).
1
Expert's answer
2020-10-02T16:00:11-0400

"\\frac{\\partial f}{\\partial x}=2xy+y(-\\sin(xy))+\\frac{1}y\\frac{1}{\\cos^2(\\frac{x}y)} \\\\\n\\frac{\\partial f}{\\partial y}=x^2+x(-\\sin(xy))+(-\\frac{x}{y^2})\\frac{1}{\\cos^2(\\frac{x}y)}\\\\\n\\frac{\\partial^2 f}{\\partial x \\partial y}=\\frac{\\partial^2 f}{\\partial y \\partial x}=2x+(-\\sin(xy))+xy(-\\cos(xy))+\\\\+(-\\frac{1}{y^2})\\frac{1}{\\cos^2(\\frac{x}y)}+(-\\frac{x}{y^2})(\\frac{1}y)2\\cos(\\frac{x}y)(-\\sin(\\frac{x}{y}))(-\\frac{1}{\\cos^4(\\frac{x}y)})=\\\\=2x-\\sin(xy)-xy\\cos(xy)-\\frac{1}{y^2}\\frac{1}{\\cos^2(\\frac{x}{y})}-\\frac{x}{y^3}\\frac{2\\sin(\\frac{x}y)}{\\cos^3(\\frac{x}y)} \\\\\n\\frac{\\partial^2 f}{\\partial x^2}=2y+y^2(-\\cos(xy))+\\frac{1}{y^2}(-\\sin(\\frac{x}{y}))(\\frac{-2}{\\cos^3(\\frac{x}{y})})=\\\\=2y-y^2\\cos(xy)+\\frac{1}{y^2}\\frac{2\\sin(\\frac{x}{y})}{\\cos^3(\\frac{x}{y})}\\\\\n\\frac{\\partial^2 f}{\\partial y^2}=-x^2\\cos(xy)+\\frac{2x}{y^3}\\frac{1}{\\cos^2(\\frac{x}y)}+\\frac{x^2}{y^4}\\frac{2\\sin(\\frac{x}y)}{\\cos^3(\\frac{x}y)}"


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