Answer to Question #249197 in Calculus for moe

Question #249197

Find the second partial derivative "\\displaystyle{\\frac{\\partial^2 z}{\\partial x^2}}" of the function "z=\\sin(xy^2)" at point "(\\pi, 1)"


1
Expert's answer
2021-10-12T04:55:45-0400

The first derivative:


"\\dfrac{\\partial}{\\partial x}(z)=\\dfrac{\\partial}{\\partial x}(\\sin(xy^2)) = y^2\\cos(xy^2)"

The second derivative:


"\\dfrac{\\partial^2z}{\\partial x^2} = \\dfrac{\\partial}{\\partial x}(y^2\\cos(xy^2)) = -y^4\\sin(xy^2)"

Substituting "(\\pi,1)", obtain:


"\\dfrac{\\partial^2z}{\\partial x^2}\\vert_{(\\pi,1)} = -1^4\\cdot \\sin(\\pi\\cdot 1^2) = 0"

Answer. 0.


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