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 (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 x^{100} d x

y = cos (5 x)

\int \frac{5}{9 + 4 x ^{2}} d x

\int_{- \infty}^{- 1} e^{- 22 t} d t

f (x) = \frac{12}{x ^{2} - 4}, (4, 4)

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