What is (\partial)/(\partial x)(xsin^2(xy-y)) ?

The answer to (\partial)/(\partial x)(xsin^2(xy-y)) is sin^2(xy-y)+xysin(2(-y+xy))

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(\partial)/(\partial x)(xsin^2(xy-y))

\frac{\partial}{\partial x} (x sin^{2} (x y - y))

sin^{2} (x y - y) + x y sin (2 (- y + x y))

Solution steps

\frac{\partial}{\partial x} (x sin^{2} (x y - y))

Treat

y

as a constant

Apply the Product Rule:

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

f = x, g = sin^{2} (x y - y)

= \frac{\partial}{\partial x} (x) sin^{2} (x y - y) + \frac{\partial}{\partial x} (sin^{2} (x y - y)) x

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

\frac{\partial}{\partial x} (sin^{2} (x y - y)) = y sin (2 (- y + y x))

= 1 \cdot sin^{2} (x y - y) + y sin (2 (- y + y x)) x

Simplify

= sin^{2} (x y - y) + x y sin (2 (- y + x y))

\int \frac{3}{4 x} d x

x^{2} - 4 x + 2

n = 0 \sum \infty \frac{1}{n ( n - 1 )}

\frac{1}{2 r} + \frac{1}{7 r ^{\frac{6}{7}}}

f (x) = \frac{x ^{2} + 11 x + 24}{x + 8}

What is (\partial)/(\partial x)(xsin^2(xy-y)) ?
The answer to (\partial)/(\partial x)(xsin^2(xy-y)) is sin^2(xy-y)+xysin(2(-y+xy))