Calculus Examples

${(\frac{1}{2})}^{4 x}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = {(\frac{1}{2})}^{x}$ and $g (x) = 4 x$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $4 x$ .

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

Step 1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $\frac{1}{2}$ .

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

Step 1.3

Replace all occurrences of $u$ with $4 x$ .

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

Step 2

Differentiate using the Constant Multiple Rule.

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Step 2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

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

Step 2.2

Simplify the expression.

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Step 2.2.1

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

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

Step 2.2.2

Rewrite $\frac{1}{2}$ as $2^{- 1}$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Multiply $- 1$ by $1$ .

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

Step 6

Multiply the exponents in ${(2^{- 1})}^{4 x}$ .

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Step 6.1

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

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

Step 6.2

Multiply $4$ by $- 1$ .

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

Step 7

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

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

Step 8

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

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

Step 9

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

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