Calculus Examples

$x^{6} e^{x}$

Step 1

Differentiate using the Product Rule which states that

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

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

$f (x) = x^{6}$ and

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

$x^{6} \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{6}]$

Step 2

Differentiate using the Exponential Rule which states that

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

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

$a$ =

$e$ .

$x^{6} e^{x} + e^{x} \frac{d}{d x} [x^{6}]$

Step 3

Differentiate using the Power Rule which states that

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

$n x^{n - 1}$ where

$n = 6$ .

$x^{6} e^{x} + e^{x} (6 x^{5})$

Step 4

Simplify.

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Step 4.1

Reorder terms.

$e^{x} x^{6} + 6 e^{x} x^{5}$

Step 4.2

Reorder factors in

$e^{x} x^{6} + 6 e^{x} x^{5}$ .

$x^{6} e^{x} + 6 x^{5} e^{x}$