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

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

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

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

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

Solution steps

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

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

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

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

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

Simplify

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

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

f^{^{'}} (x) = x^{2} - x, f (3) = \frac{19}{2}

\frac{\partial}{\partial x} (\frac{x}{x + 4})

\int 6 cos (3 x) d x

x \to 0 lim (\frac{sin ( 10 x )}{sin ( 3 x )})

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

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