What is (\partial)/(\partial x)(xcos(4xy)) ?

The answer to (\partial)/(\partial x)(xcos(4xy)) is cos(4xy)-4xysin(4xy)

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(\partial)/(\partial x)(xcos(4xy))

\frac{\partial}{\partial x} (x cos (4 x y))

cos (4 x y) - 4 x y sin (4 x y)

Solution steps

\frac{\partial}{\partial x} (x cos (4 x y))

Treat

y

as a constant

Apply the Product Rule:

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

f = x, g = cos (4 x y)

= \frac{\partial}{\partial x} (x) cos (4 x y) + \frac{\partial}{\partial x} (cos (4 x y)) x

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

\frac{\partial}{\partial x} (cos (4 x y)) = - sin (4 x y) \cdot 4 y

= 1 \cdot cos (4 x y) + (- sin (4 x y) \cdot 4 y) x

Simplify

= cos (4 x y) - 4 x y sin (4 x y)

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What is (\partial)/(\partial x)(xcos(4xy)) ?
The answer to (\partial)/(\partial x)(xcos(4xy)) is cos(4xy)-4xysin(4xy)