What is (\partial)/(\partial x)(-xsin(xy)) ?

The answer to (\partial)/(\partial x)(-xsin(xy)) is -sin(xy)-xycos(xy)

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(\partial)/(\partial x)(-xsin(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

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

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

- (1 \cdot sin (x y) + cos (x y) y x) : - sin (x y) - x y cos (x y)

= - sin (x y) - x y cos (x y)

\frac{d}{d x} (x \cdot e^{x} \cdot sin (x))

\int_{8}^{10} \frac{y}{y ^{2} - 4 y - 5} d y

f (x) = (- 2 x^{3} + x^{2} - 2 x - 1) x^{3}

y^{^{''}} + 64 y = sin (8 t)

x \to 0 lim (x^{2} + 4)

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