differentiate x^3*e^x — step-by-step solution

Let's differentiate:

\frac{d}{d x} (x^{3} e^{x})

For

u1

x^{3}

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)

Let's perform the differentiation

u1

\frac{d}{d x} (x^{3})

ⓘ The derivative of

x^{n}

n x^{n - 1}

\frac{d}{d x} (x^{3})

3 x^{2}

3 x^{2}

Let's perform the differentiation

v2

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

ⓘ Differentiating

\exp (x)

gives

\exp (x)

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

e^{x}

e^{x}

Then you can resume the differentiation:

ⅰ

\frac{d}{d x} (x^{3} e^{x})

3 x^{2} e^{x} + x^{3} e^{x}

3 x^{2} e^{x} + x^{3} e^{x}

e^{x} x^{3} + 3 e^{x} x^{2}

e^{x} x^{2} \cdot (x + 3)

▶Derivative:

e^{x} x^{2} \cdot (x + 3)