In Exercises 9 – 26, find fx, fy, fxx, fyy, fxy and fyx.
Solution:
To find fx, you take the derivative of f(x, y) and treat y as a constant.
fx(x, y) = \(-5{ y }^{ 3 }sin(5x{ y }^{ 3 })\)
To find fy, you take the derivative of f(x, y) and treat x as a constant.
fy(x, y) = \(-15x{ y }^{ 2 }sin(5x{ y }^{ 3 })\)
To find fxx, you take the derivative of fx and treat y as a constant.
fxx(x, y) = \(-25{ y }^{ 6 }cos(5x{ y }^{ 3 })\)
To find fyy, you take the derivative of fy and treat x as a constant.
\(-15x{ y }^{ 2 }sin(5x{ y }^{ 3 })\) Take out constants.
\(-15x({ y }^{ 2 }sin(5x{ y }^{ 3 }))\) Take the derivative with regards to derivative of products.
fyy(x, y) = \(-15x(2ysin(5x{ y }^{ 3 })+15x{ y }^{ 4 }cos(5x{ y }^{ 3 }))\)
To find fxy, you take the derivative of fx and treat x as a constant.
\(-5{ y }^{ 3 }sin(5x{ y }^{ 3 })\) Take out constant.
\(-5({ y }^{ 3 }sin(5x{ y }^{ 3 }))\) Take the derivative with regards to derivative of products.
fxy(x, y) = \(-5(3{ y }^{ 2 }sin(5x{ y }^{ 3 })+15x{ y }^{ 5 }cos(5x{ y }^{ 3 }))\)
To find fyx, you take the derivative of fy and treat y as a constant.
\(-15x{ y }^{ 2 }sin(5x{ y }^{ 3 })\) Take out constants.
\(-15{ y }^{ 2 }(xsin(5x{ y }^{ 3 }))\) Take the derivative with regards to derivative of products.
\(-15{ y }^{ 2 }(sin(5x{ y }^{ 3 })5x{ y }^{ 3 }+cos(5x{ y }^{ 3 }))\)