Calculus Examples

y = x^{3} e^{x}

$y = x^{3} e^{x}$

Step 1

Differentiate using the Product Rule which states that

\frac{d}{d x} [f (x) g (x)]

$\frac{d}{d x} [f (x) g (x)]$ is

f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]

$f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where

f (x) = x^{3}

$f (x) = x^{3}$ and

g (x) = e^{x}

$g (x) = e^{x}$ .

x^{3} \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{3}]

$x^{3} \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{3}]$

Step 2

Differentiate using the Exponential Rule which states that

\frac{d}{d x} [a^{x}]

$\frac{d}{d x} [a^{x}]$ is

a^{x} \ln (a)

$a^{x} \ln (a)$ where

a

$a$ =

e

$e$ .

x^{3} e^{x} + e^{x} \frac{d}{d x} [x^{3}]

$x^{3} e^{x} + e^{x} \frac{d}{d x} [x^{3}]$

Step 3

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 3

$n = 3$ .

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

$x^{3} e^{x} + e^{x} (3 x^{2})$

Step 4

Simplify.

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Step 4.1

Reorder terms.

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

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

Step 4.2

Reorder factors in

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

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

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

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

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

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