What is (\partial)/(\partial x)(xsin(8x^2y)) ?

The answer to (\partial)/(\partial x)(xsin(8x^2y)) is sin(8x^2y)+16x^2ycos(8x^2y)

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(\partial)/(\partial x)(xsin(8x^2y))

\frac{\partial}{\partial x} (x sin (8 x^{2} y))

sin (8 x^{2} y) + 16 x^{2} y cos (8 x^{2} y)

Solution steps

\frac{\partial}{\partial x} (x sin (8 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 (8 x^{2} y)

= \frac{\partial}{\partial x} (x) sin (8 x^{2} y) + \frac{\partial}{\partial x} (sin (8 x^{2} y)) x

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

\frac{\partial}{\partial x} (sin (8 x^{2} y)) = cos (8 x^{2} y) \cdot 16 y x

= 1 \cdot sin (8 x^{2} y) + cos (8 x^{2} y) \cdot 16 y xx

Simplify

1 \cdot sin (8 x^{2} y) + cos (8 x^{2} y) \cdot 16 y xx : sin (8 x^{2} y) + 16 x^{2} y cos (8 x^{2} y)

= sin (8 x^{2} y) + 16 x^{2} y cos (8 x^{2} y)

\frac{- 60}{- 2 + 5 s}

(\frac{e ^{x} + e ^{- x}}{2})^{^{'}}

y = x^{\frac{1}{3}}

t \to 1 lim (ln (t - 1))

\frac{d}{d x} ((\frac{x ^{2}}{4} + 5 x - \frac{1}{x})^{3})

What is (\partial)/(\partial x)(xsin(8x^2y)) ?
The answer to (\partial)/(\partial x)(xsin(8x^2y)) is sin(8x^2y)+16x^2ycos(8x^2y)