What is (\partial)/(\partial x)(xcos(x^2y)) ?

The answer to (\partial)/(\partial x)(xcos(x^2y)) is cos(x^2y)-2x^2ysin(x^2y)

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(\partial)/(\partial x)(xcos(x^2y))

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

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

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

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

What is (\partial)/(\partial x)(xcos(x^2y)) ?
The answer to (\partial)/(\partial x)(xcos(x^2y)) is cos(x^2y)-2x^2ysin(x^2y)