Precalculus Examples

$f (x) = 2^{x^{9}}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = 2^{x}$ and $g (x) = x^{9}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $x^{9}$ .

$\frac{d}{d u} [2^{u}] \frac{d}{d x} [x^{9}]$

Step 1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $2$ .

$2^{u} \ln (2) \frac{d}{d x} [x^{9}]$

Step 1.3

Replace all occurrences of $u$ with $x^{9}$ .

$2^{x^{9}} \ln (2) \frac{d}{d x} [x^{9}]$

Step 2

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

$2^{x^{9}} \ln (2) (9 x^{8})$

Step 3

Simplify.

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Step 3.1

Reorder the factors of $2^{x^{9}} \ln (2) (9 x^{8})$ .

$9 \cdot 2^{x^{9}} x^{8} \ln (2)$

Step 3.2

Reorder factors in $9 \cdot 2^{x^{9}} x^{8} \ln (2)$ .

$9 x^{8} \cdot 2^{x^{9}} \ln (2)$