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)

\int x^{2} x^{3} + 21 d x

\int + \frac{x ^{2}}{( 225 + x ^{2} ) ^{2}} d x

x \to 0 lim (\frac{( 3 ^{x} - 4 ^{x} )}{x})

e^{9 x}

\frac{1}{3} sin (3 x)

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