What is (\partial)/(\partial x)(x*sin(xy)) ?

The answer to (\partial)/(\partial x)(x*sin(xy)) is sin(xy)+xycos(xy)

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(\partial)/(\partial x)(x*sin(xy))

\frac{\partial}{\partial x} (x \cdot sin (x y))

sin (x y) + x y cos (x y)

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

f = x, g = sin (x y)

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

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

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

= 1 \cdot sin (x y) + cos (x y) y x

Simplify

= sin (x y) + x y cos (x y)

\int 5 x^{\frac{1}{3}} + 4 x^{- \frac{1}{3}} + 11 d x

\frac{d}{d x} (\frac{( x ) ^{3}}{( 2 x ^{7} + x ) ^{2}})

\frac{d}{d x} ((x^{2} + 2)^{5})

x \to 4 - lim (\frac{x - 4}{∣ x - 4 ∣})

y^{2} + 2 x + 2 x y y^{^{'}} = 0

What is (\partial)/(\partial x)(x*sin(xy)) ?
The answer to (\partial)/(\partial x)(x*sin(xy)) is sin(xy)+xycos(xy)