Calculus Examples

${(\frac{1}{81})}^{2 x}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = {(\frac{1}{81})}^{x}$ and $g (x) = 2 x$ .

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Step 1.1

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

$\frac{d}{d u} [{(\frac{1}{81})}^{u}] \frac{d}{d x} [2 x]$

Step 1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $\frac{1}{81}$ .

${(\frac{1}{81})}^{u} \ln (\frac{1}{81}) \frac{d}{d x} [2 x]$

Step 1.3

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

${(\frac{1}{81})}^{2 x} \ln (\frac{1}{81}) \frac{d}{d x} [2 x]$

Step 2

Differentiate.

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Step 2.1

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

${(\frac{1}{81})}^{2 x} \ln (\frac{1}{81}) (2 \frac{d}{d x} [x])$

Step 2.2

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

${(\frac{1}{81})}^{2 x} \ln (\frac{1}{81}) (2 \cdot 1)$

Step 2.3

Simplify the expression.

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Step 2.3.1

Multiply $2$ by $1$ .

${(\frac{1}{81})}^{2 x} \ln (\frac{1}{81}) \cdot 2$

Step 2.3.2

Move $2$ to the left of ${(\frac{1}{81})}^{2 x} \ln (\frac{1}{81})$ .

$2 \cdot ({(\frac{1}{81})}^{2 x} \ln (\frac{1}{81}))$

Step 3

Simplify.

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Step 3.1

Apply the product rule to $\frac{1}{81}$ .

$2 \frac{1^{2 x}}{81^{2 x}} \ln (\frac{1}{81})$

Step 3.2

Combine terms.

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Step 3.2.1

One to any power is one.

$2 \frac{1}{81^{2 x}} \ln (\frac{1}{81})$

Step 3.2.2

Combine $2$ and $\frac{1}{81^{2 x}}$ .

$\frac{2}{81^{2 x}} \ln (\frac{1}{81})$

Step 3.2.3

Combine $\frac{2}{81^{2 x}}$ and $\ln (\frac{1}{81})$ .

$\frac{2 \ln (\frac{1}{81})}{81^{2 x}}$