What is (\partial)/(\partial y)(ysin(xy)) ?

The answer to (\partial)/(\partial y)(ysin(xy)) is sin(yx)+yxcos(yx)

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(\partial)/(\partial y)(ysin(xy))

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

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

Solution steps

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

Treat

x

as a constant

Apply the Product Rule:

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

f = y, g = sin (x y)

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

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

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

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

Simplify

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

\frac{d}{d x} ((ln (x))^{5})

\int 8 x e^{8 x} d x

ln^{3} (x)

\frac{d y}{d x} = \frac{( 1 - 3 y ) ^{2}}{x}, y (1) = 1

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

What is (\partial)/(\partial y)(ysin(xy)) ?
The answer to (\partial)/(\partial y)(ysin(xy)) is sin(yx)+yxcos(yx)