What is (\partial)/(\partial x)(xsin(4x^2y)) ?

The answer to (\partial)/(\partial x)(xsin(4x^2y)) is sin(4x^2y)+8x^2ycos(4x^2y)

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(\partial)/(\partial x)(xsin(4x^2y))

\frac{\partial}{\partial x} (x sin (4 x^{2} y))

sin (4 x^{2} y) + 8 x^{2} y cos (4 x^{2} y)

Solution steps

\frac{\partial}{\partial x} (x sin (4 x^{2} y))

Treat

y

as a constant

Apply the Product Rule:

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

f = x, g = sin (4 x^{2} y)

= \frac{\partial}{\partial x} (x) sin (4 x^{2} y) + \frac{\partial}{\partial x} (sin (4 x^{2} y)) x

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

\frac{\partial}{\partial x} (sin (4 x^{2} y)) = cos (4 x^{2} y) \cdot 8 y x

= 1 \cdot sin (4 x^{2} y) + cos (4 x^{2} y) \cdot 8 y xx

Simplify

1 \cdot sin (4 x^{2} y) + cos (4 x^{2} y) \cdot 8 y xx : sin (4 x^{2} y) + 8 x^{2} y cos (4 x^{2} y)

= sin (4 x^{2} y) + 8 x^{2} y cos (4 x^{2} y)

What is (\partial)/(\partial x)(xsin(4x^2y)) ?
The answer to (\partial)/(\partial x)(xsin(4x^2y)) is sin(4x^2y)+8x^2ycos(4x^2y)