Calculus Examples

x^{9} e^{x}

$x^{9} e^{x}$

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^{9}

$f (x) = x^{9}$ and

g (x) = e^{x}

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

x^{9} \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{9}]

$x^{9} \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{9}]$

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^{9} e^{x} + e^{x} \frac{d}{d x} [x^{9}]

$x^{9} e^{x} + e^{x} \frac{d}{d x} [x^{9}]$

Step 3

Differentiate using the Power Rule which states that

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

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

$n x^{n - 1}$ where

$n = 9$ .

$x^{9} e^{x} + e^{x} (9 x^{8})$

Step 4

Simplify.

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Step 4.1

Reorder terms.

$e^{x} x^{9} + 9 e^{x} x^{8}$

Step 4.2

Reorder factors in

$e^{x} x^{9} + 9 e^{x} x^{8}$ .

$x^{9} e^{x} + 9 x^{8} e^{x}$