What is (\partial)/(\partial x)(e^{x+y}sin(x-y)) ?

The answer to (\partial)/(\partial x)(e^{x+y}sin(x-y)) is e^{x+y}sin(x-y)+cos(x-y)e^{x+y}

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(\partial)/(\partial x)(e^{x+y}sin(x-y))

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

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

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

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

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

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

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

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

\frac{d}{d x} (\frac{2}{3} (e^{- 2 x} + e^{x}))

\frac{x ^{4}}{2 - x ^{3}}

\frac{d}{d x} (47 x^{3} - 75 x)

y^{^{''}} + 7 y^{^{'}} + 10 y = 10 t e^{4 t}

\int_{0}^{1} \frac{57}{x ^{5}} d x

What is (\partial)/(\partial x)(e^{x+y}sin(x-y)) ?
The answer to (\partial)/(\partial x)(e^{x+y}sin(x-y)) is e^{x+y}sin(x-y)+cos(x-y)e^{x+y}