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

The answer to (\partial)/(\partial y)(e^ysin(xy)) is e^ysin(xy)+cos(xy)xe^y

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

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

e^{y} sin (x y) + cos (x y) x e^{y}

Solution steps

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

Treat

x

as a constant

Apply the Product Rule:

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

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

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

\frac{\partial}{\partial y} (e^{y}) = e^{y}

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

= e^{y} sin (x y) + cos (x y) x e^{y}

\int \frac{9 x}{( 2 - x ^{2} ) ^{3}} d x

\frac{\partial}{\partial x} (x^{3} sin (x y))

\int (1 - e^{- 10 t} - 10 t e^{- 10 t}) d t

y = 3 x, y = 3, x = 0

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

What is (\partial)/(\partial y)(e^ysin(xy)) ?
The answer to (\partial)/(\partial y)(e^ysin(xy)) is e^ysin(xy)+cos(xy)xe^y