Calculus Examples

$2^{2^{x}}$

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 $g (x) = 2^{x}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $2^{x}$ .

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $2^{x}$ .

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

Step 2

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

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

Step 3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2^{x + 2^{x}} \ln (2) \ln (2)$

Step 4

Raise $\ln (2)$ to the power of $1$ .

$2^{x + 2^{x}} (\ln^{1} (2) \ln (2))$

Step 5

Raise $\ln (2)$ to the power of $1$ .

$2^{x + 2^{x}} (\ln^{1} (2) \ln^{1} (2))$

Step 6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2^{x + 2^{x}} {\ln (2)}^{1 + 1}$

Step 7

Add $1$ and $1$ .

$2^{x + 2^{x}} \ln^{2} (2)$