What is the derivative of x*e^y ?

The derivative of x*e^y is e^y+e^yx(dy)/(dx)

What is the first derivative of x*e^y ?

The first derivative of x*e^y is e^y+e^yx(dy)/(dx)

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

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

Solution steps

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

Apply the Product Rule:

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

f = x, g = e^{y}

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

\frac{d x}{d x} = 1

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

= 1 \cdot e^{y} + e^{y} \frac{d y}{d x} x

Simplify

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

What is the derivative of x*e^y ?
The derivative of x*e^y is e^y+e^yx(dy)/(dx)
What is the first derivative of x*e^y ?
The first derivative of x*e^y is e^y+e^yx(dy)/(dx)