Calculus Examples

$4^{x^{2}}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = 4^{x}$ and $g (x) = x^{2}$ .

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Step 1.1

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

$\frac{d}{d u} [4^{u}] \frac{d}{d x} [x^{2}]$

Step 1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $4$ .

$4^{u} \ln (4) \frac{d}{d x} [x^{2}]$

Step 1.3

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

$4^{x^{2}} \ln (4) \frac{d}{d x} [x^{2}]$

Step 2

Differentiate using the Power Rule.

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Step 2.1

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

$4^{x^{2}} \ln (4) (2 x)$

Step 2.2

Simplify the expression.

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Step 2.2.1

Rewrite $4$ as $2^{2}$ .

$2 \cdot {(2^{2})}^{x^{2}} \ln (4) x$

Step 2.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2 \cdot 2^{2 x^{2}} \ln (4) x$

Step 3

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

$2^{1 + 2 x^{2}} \ln (4) x$

Step 4

Reorder the factors of $2^{1 + 2 x^{2}} \ln (4) x$ .

$2^{1 + 2 x^{2}} x \ln (4)$