Calculus Examples

$e^{\frac{1}{9 x}}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = \frac{1}{9 x}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{1}{9 x}$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d x} [\frac{1}{9 x}]$

Step 1.2

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

$e^{u} \frac{d}{d x} [\frac{1}{9 x}]$

Step 1.3

Replace all occurrences of $u$ with $\frac{1}{9 x}$ .

$e^{\frac{1}{9 x}} \frac{d}{d x} [\frac{1}{9 x}]$

Step 2

Differentiate.

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Step 2.1

Since $\frac{1}{9}$ is constant with respect to $x$ , the derivative of $\frac{1}{9 x}$ with respect to $x$ is $\frac{1}{9} \frac{d}{d x} [\frac{1}{x}]$ .

$e^{\frac{1}{9 x}} (\frac{1}{9} \frac{d}{d x} [\frac{1}{x}])$

Step 2.2

Combine fractions.

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Step 2.2.1

Combine $\frac{1}{9}$ and $e^{\frac{1}{9 x}}$ .

$\frac{e^{\frac{1}{9 x}}}{9} \frac{d}{d x} [\frac{1}{x}]$

Step 2.2.2

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

$\frac{e^{\frac{1}{9 x}}}{9} \frac{d}{d x} [x^{- 1}]$

Step 2.3

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

$\frac{e^{\frac{1}{9 x}}}{9} (- x^{- 2})$

Step 2.4

Combine fractions.

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Step 2.4.1

Combine $\frac{e^{\frac{1}{9 x}}}{9}$ and $x^{- 2}$ .

$- \frac{e^{\frac{1}{9 x}} x^{- 2}}{9}$

Step 2.4.2

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{e^{\frac{1}{9 x}}}{9 x^{2}}$