Calculus Examples

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

Step 1

By the Sum Rule, the derivative of $e^{x} x^{9} + 9 e^{x} x^{8}$ with respect to $x$ is $\frac{d}{d x} [e^{x} x^{9}] + \frac{d}{d x} [9 e^{x} x^{8}]$ .

$\frac{d}{d x} [e^{x} x^{9}] + \frac{d}{d x} [9 e^{x} x^{8}]$

Step 2

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

Tap for more steps...

Step 2.1

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 $f (x) = e^{x}$ and $g (x) = x^{9}$ .

$e^{x} \frac{d}{d x} [x^{9}] + x^{9} \frac{d}{d x} [e^{x}] + \frac{d}{d x} [9 e^{x} x^{8}]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 9$ .

$e^{x} (9 x^{8}) + x^{9} \frac{d}{d x} [e^{x}] + \frac{d}{d x} [9 e^{x} x^{8}]$

Step 2.3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$e^{x} (9 x^{8}) + x^{9} e^{x} + \frac{d}{d x} [9 e^{x} x^{8}]$

Step 3

Evaluate $\frac{d}{d x} [9 e^{x} x^{8}]$ .

Tap for more steps...

Step 3.1

Since $9$ is constant with respect to $x$ , the derivative of $9 e^{x} x^{8}$ with respect to $x$ is $9 \frac{d}{d x} [e^{x} x^{8}]$ .

$e^{x} (9 x^{8}) + x^{9} e^{x} + 9 \frac{d}{d x} [e^{x} x^{8}]$

Step 3.2

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 $f (x) = e^{x}$ and $g (x) = x^{8}$ .

$e^{x} (9 x^{8}) + x^{9} e^{x} + 9 (e^{x} \frac{d}{d x} [x^{8}] + x^{8} \frac{d}{d x} [e^{x}])$

Step 3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 8$ .

$e^{x} (9 x^{8}) + x^{9} e^{x} + 9 (e^{x} (8 x^{7}) + x^{8} \frac{d}{d x} [e^{x}])$

Step 3.4

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$e^{x} (9 x^{8}) + x^{9} e^{x} + 9 (e^{x} (8 x^{7}) + x^{8} e^{x})$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Apply the distributive property.

$e^{x} (9 x^{8}) + x^{9} e^{x} + 9 (e^{x} (8 x^{7})) + 9 (x^{8} e^{x})$

Step 4.2

Combine terms.

Tap for more steps...

Step 4.2.1

Multiply $8$ by $9$ .

$e^{x} (9 x^{8}) + x^{9} e^{x} + 72 (e^{x} (x^{7})) + 9 (x^{8} e^{x})$

Step 4.2.2

Add $e^{x} (9 x^{8})$ and $9 x^{8} e^{x}$ .

Tap for more steps...

Step 4.2.2.1

Move $e^{x}$ .

$9 x^{8} e^{x} + 9 x^{8} e^{x} + x^{9} e^{x} + 72 e^{x} x^{7}$

Step 4.2.2.2

Add $9 x^{8} e^{x}$ and $9 x^{8} e^{x}$ .

$18 x^{8} e^{x} + x^{9} e^{x} + 72 e^{x} x^{7}$

Step 4.3

Reorder terms.

$18 e^{x} x^{8} + e^{x} x^{9} + 72 e^{x} x^{7}$

Step 4.4

Reorder factors in $18 e^{x} x^{8} + e^{x} x^{9} + 72 e^{x} x^{7}$ .

$18 x^{8} e^{x} + x^{9} e^{x} + 72 x^{7} e^{x}$