What is (\partial)/(\partial x)(xsin(7x^2y)) ?

The answer to (\partial)/(\partial x)(xsin(7x^2y)) is sin(7x^2y)+14x^2ycos(7x^2y)

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(\partial)/(\partial x)(xsin(7x^2y))

\frac{\partial}{\partial x} (x sin (7 x^{2} y))

sin (7 x^{2} y) + 14 x^{2} y cos (7 x^{2} y)

Solution steps

\frac{\partial}{\partial x} (x sin (7 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 (7 x^{2} y)

= \frac{\partial}{\partial x} (x) sin (7 x^{2} y) + \frac{\partial}{\partial x} (sin (7 x^{2} y)) x

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

\frac{\partial}{\partial x} (sin (7 x^{2} y)) = cos (7 x^{2} y) \cdot 14 y x

= 1 \cdot sin (7 x^{2} y) + cos (7 x^{2} y) \cdot 14 y xx

Simplify

1 \cdot sin (7 x^{2} y) + cos (7 x^{2} y) \cdot 14 y xx : sin (7 x^{2} y) + 14 x^{2} y cos (7 x^{2} y)

= sin (7 x^{2} y) + 14 x^{2} y cos (7 x^{2} y)

x \to 0 lim (\frac{x ^{2} + 1}{x})

G (t) = \frac{1 - 5 t}{4 + t}

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

\int e^{2 t} d t

\int \frac{9 - x ^{2}}{2 x ^{2}} d x

What is (\partial)/(\partial x)(xsin(7x^2y)) ?
The answer to (\partial)/(\partial x)(xsin(7x^2y)) is sin(7x^2y)+14x^2ycos(7x^2y)