What is (\partial)/(\partial x)(-10xysin(xy^2)) ?

The answer to (\partial)/(\partial x)(-10xysin(xy^2)) is -10y(sin(xy^2)+xy^2cos(xy^2))

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(\partial)/(\partial x)(-10xysin(xy^2))

\frac{\partial}{\partial x} (- 10 x y sin (x y^{2}))

- 10 y (sin (x y^{2}) + x y^{2} cos (x y^{2}))

Solution steps

\frac{\partial}{\partial x} (- 10 x y sin (x y^{2}))

Treat

y

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= - 10 y \frac{\partial}{\partial x} (x sin (x y^{2}))

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x, g = sin (x y^{2})

= - 10 y (\frac{\partial}{\partial x} (x) sin (x y^{2}) + \frac{\partial}{\partial x} (sin (x y^{2})) x)

\frac{\partial}{\partial x} (x) = 1

\frac{\partial}{\partial x} (sin (x y^{2})) = cos (x y^{2}) y^{2}

= - 10 y (1 \cdot sin (x y^{2}) + cos (x y^{2}) y^{2} x)

Simplify

= - 10 y (sin (x y^{2}) + x y^{2} cos (x y^{2}))

What is (\partial)/(\partial x)(-10xysin(xy^2)) ?
The answer to (\partial)/(\partial x)(-10xysin(xy^2)) is -10y(sin(xy^2)+xy^2cos(xy^2))