Calculus Examples

$f (x) = \ln (\frac{e}{x^{n}})$

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{e}{x^{n}}$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [\frac{e}{x^{n}}]$

Step 1.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

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

Step 1.3

Replace all occurrences of $u$ with $\frac{e}{x^{n}}$ .

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

Step 2

Multiply by the reciprocal of the fraction to divide by $\frac{e}{x^{n}}$ .

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

Step 3

Multiply $\frac{x^{n}}{e}$ by $1$ .

$\frac{x^{n}}{e} \frac{d}{d x} [\frac{e}{x^{n}}]$

Step 4

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

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

Step 5

Simplify terms.

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Step 5.1

Combine $e$ and $\frac{x^{n}}{e}$ .

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

Step 5.2

Cancel the common factor of $e$ .

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Step 5.2.1

Cancel the common factor.

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

Step 5.2.2

Divide $x^{n}$ by $1$ .

$x^{n} \frac{d}{d x} [\frac{1}{x^{n}}]$

Step 5.3

Apply basic rules of exponents.

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Step 5.3.1

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

$x^{n} \frac{d}{d x} [{(x^{n})}^{- 1}]$

Step 5.3.2

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

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Step 5.3.2.1

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

$x^{n} \frac{d}{d x} [x^{n \cdot - 1}]$

Step 5.3.2.2

Move $- 1$ to the left of $n$ .

$x^{n} \frac{d}{d x} [x^{- 1 \cdot n}]$

Step 5.3.2.3

Rewrite $- 1 n$ as $- n$ .

$x^{n} \frac{d}{d x} [x^{- n}]$

Step 6

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

$x^{n} (- n x^{- n - 1})$

Step 7

Multiply $x^{n}$ by $x^{- n - 1}$ by adding the exponents.

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Step 7.1

Move $x^{- n - 1}$ .

$x^{- n - 1} x^{n} (- n)$

Step 7.2

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

$x^{- n - 1 + n} (- n)$

Step 7.3

Combine the opposite terms in $- n - 1 + n$ .

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Step 7.3.1

Add $- n$ and $n$ .

$x^{0 - 1} (- n)$

Step 7.3.2

Subtract $1$ from $0$ .

$x^{- 1} (- n)$

Step 8

Simplify.

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Step 8.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{x} (- n)$

Step 8.2

Combine $\frac{1}{x}$ and $n$ .

$- \frac{n}{x}$