What is (\partial)/(\partial x)(xysin(xy)) ?

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

(\partial)/(\partial x)(xysin(xy))

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

= y \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)

= 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

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

Simplify

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

\int \frac{e ^{x}}{4 + e ^{x}} d x

\frac{d}{d t} (t^{2} + 1)

\frac{d y}{d t} = \frac{t ^{2}}{y ( 1 + t ^{3} )}

\int_{3}^{4} \frac{x}{8} d x

\frac{\partial}{\partial r} (π r^{2} h)

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