What is the d/(dy)((x+y)e^y) ?

The d/(dy)((x+y)e^y) is e^y+e^y(x+y)

What is the first d/(dy)((x+y)e^y) ?

The first d/(dy)((x+y)e^y) is e^y+e^y(x+y)

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\frac{d}{d y} ((x + y) e^{y})

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

Solution steps

\frac{d}{d y} ((x + y) e^{y})

Treat

x

as a constant

Apply the Product Rule:

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

f = x + y, g = e^{y}

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

\frac{d}{d y} (x + y) = 1

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

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

Simplify

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

\frac{d}{d x} ((4 + 8 x) cos (1 - x))

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\int \frac{4 x + 1}{2 x ^{2} + x - 3} d x

\frac{4}{s ( s ^{2} + 4 )}

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