Calculus Examples

$2^{x^{2}}$

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Reorder the factors of $2^{1 + x^{2}} \ln (2) x$ .

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