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

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 3

Raise $2$ to the power of $1$ .

$2^{1} \cdot 2^{2 x} \ln (2) \frac{d}{d x} [x]$

Step 4

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

$2^{1 + 2 x} \ln (2) \frac{d}{d x} [x]$

Step 5

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

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

Step 6

Multiply $2^{1 + 2 x}$ by $1$ .

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