differentiate x*e^x — step-by-step solution

1) Let's perform the differentiation:

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

For

u1

x

and

v2

e^{x}

, use the differentiation rule:

\frac{d}{d x} (u1 \cdot v2) = \frac{d}{d x} (u1) \cdot v2 + u1 \cdot \frac{d}{d x} (v2)

2) Let's differentiate

u1

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

ⓘ The derivative of x is 1:

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

1

1

3) Let's perform the differentiation

v2

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

ⓘ The derivative of exp(x) is exp(x):

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

e^{x}

e^{x}

4) Let's evaluate the resulting derivative (

u1 \cdot v2

)':

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

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

5) Then you can continue the derivation:

ⅰ

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

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

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

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

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

▶Derivative:

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