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)

y^{8} - \frac{x ^{4} y ^{9} + 4}{x ^{5} + 4 y ^{^{'}}} = 0

d er i v a t i v e \frac{3 x - 2}{2 x + 1}

d er i v a t i v e f (x) = e^{t a n (θ)}

\frac{d y}{d θ} = sin (θ)

\int x e^{x y} + 2 x y + \frac{1}{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)