Question

Find all the second-order partial derivatives of the functions g(x,y)=\cos x^{2}-\sin 3y

Derivatives
ANSWERED
asked 2021-05-18
Find all the second-order partial derivatives of the functions \(\displaystyle{g{{\left({x},{y}\right)}}}={{\cos{{x}}}^{{{2}}}-}{\sin{{3}}}{y}\)

Answers (1)

2021-05-19
Step 1: Given that
Find all the second-order partial derivatives of the functions \(\displaystyle{g{{\left({x},{y}\right)}}}={{\cos{{x}}}^{{{2}}}-}{\sin{{3}}}{y}\)
Step 2: Finding the Second Order Partial Derivative
\(\displaystyle{g{{\left({x},{y}\right)}}}={{\cos{{x}}}^{{{2}}}-}{\sin{{3}}}{y}\)
\(\displaystyle{\frac{{\partial}}{{\partial{x}}}}{\left({g{{\left({x},{y}\right)}}}\right)}={\frac{{\partial}}{{\partial{x}}}}{\left({{\cos{{x}}}^{{{2}}}-}{\sin{{3}}}{y}\right)}=-{\sin{{\left({x}^{{{2}}}\right)}}}\times{2}{x}-{0}=-{2}{x}{\sin{{\left({x}^{{{2}}}\right)}}}\)
\(\displaystyle{\frac{{\partial^{{{2}}}}}{{\partial{x}^{{{2}}}}}}{\left({g{{\left({x},{y}\right)}}}\right)}={\frac{{\partial}}{{\partial{y}}}}{\left({{\cos{{x}}}^{{{2}}}-}{\sin{{3}}}{y}\right)}=-{3}{\cos{{3}}}{y}\)
\(\displaystyle{\frac{{\partial^{{{2}}}}}{{\partial{y}^{{{2}}}}}}{\left({g{{\left({x},{y}\right)}}}\right)}={\frac{{\partial}}{{\partial{x}}}}{\left(-{3}{\cos{{3}}}{y}\right)}={9}{\sin{{3}}}{y}\)
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