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

1) Let's differentiate:

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

We differentiate

u1

(1 + x)

and

v2

e^{x}

using:

\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} (1 + x)

ⓘ The derivative of a sum is the sum of the derivatives:

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

0 + 1

0 + 1

ⓘ Adding 0 changes nothing:

0 + 1

1

1

3) Let's differentiate

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) We evaluate the resulting derivative (

u1 \cdot v2

)':

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

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

ⓘ Expand by multiplying each term inside the parentheses:

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

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

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

ⓘ Like terms are grouped by adding their coefficients:

e^{x} + e^{x}

2 e^{x}

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

5) You can then resume the derivation:

ⅰ

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

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

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

e^{x} x + 2 e^{x}

e^{x} \cdot (x + 2)

▶Derivative:

e^{x} \cdot (x + 2)