Calculus Examples

$4^{2 x}$

Step 1

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

Tap for more steps...

Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $2 x$ .

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

Step 2

Differentiate using the Constant Multiple Rule.

Tap for more steps...

Step 2.1

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

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

Step 2.2

Simplify the expression.

Tap for more steps...

Step 2.2.1

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

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

Step 2.2.2

Multiply the exponents in ${(2^{2})}^{2 x}$ .

Tap for more steps...

Step 2.2.2.1

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

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

Step 2.2.2.2

Multiply $2$ by $2$ .

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

Step 3

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

$2^{1 + 4 x} \ln (4) \frac{d}{d x} [x]$

Step 4

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

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

Step 5

Multiply $2^{1 + 4 x}$ by $1$ .

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