What is (\partial)/(\partial y)(ye^{xy}sin(y)) ?

The answer to (\partial)/(\partial y)(ye^{xy}sin(y)) is e^{yx}sin(y)+y(e^{yx}xsin(y)+e^{yx}cos(y))

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(\partial)/(\partial y)(ye^{xy}sin(y))

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

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

Solution steps

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

Treat

x

as a constant

Apply the Product Rule:

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

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

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

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

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

= 1 \cdot e^{x y} sin (y) + (e^{x y} x sin (y) + cos (y) e^{x y}) y

Simplify

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

x \to 2 - lim (3 x)

x \to 0 + lim (x^{3} cot (x))

5 x^{^{''}} + 50 x - 10 = 0

\frac{\partial}{\partial y} ((2 x + 3 y)^{7})

\frac{d}{d u} (ln (u cos (v)))

What is (\partial)/(\partial y)(ye^{xy}sin(y)) ?
The answer to (\partial)/(\partial y)(ye^{xy}sin(y)) is e^{yx}sin(y)+y(e^{yx}xsin(y)+e^{yx}cos(y))