derivative of (1+xe^x) — step-by-step solution

⚠ xe is read as the product x*e because x appears elsewhere

⚠ xe is read as the product x*e, where e is the exponential constant

Let's differentiate:

\frac{d}{d x} (1 + x e^{x})

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

For

U1

x

and

V2

e^{x}

, use the product/quotient rule:

\frac{d}{d x} (U1 \cdot V2) = \frac{d}{d x} (U1) \cdot V2 + U1 \cdot \frac{d}{d x} (V2)

Let's perform the differentiation

U1

\frac{d}{d x} (x)

1

Let's perform the differentiation

V2

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

e^{x}

Let's evaluate the resulting derivative (

U1 \cdot V2

)':

1 \cdot e^{x} + x e^{x}

e^{x} + x e^{x}

e^{x} x + e^{x}

▶You can factorize this expression into:

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