Calculus Examples

$\ln (\frac{e}{e - x})$

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\frac{e}{e - x}$ .

$\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [\frac{e}{e - x}]$

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u_{1}$ with $\frac{e}{e - x}$ .

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

Step 2

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

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

Step 3

Differentiate using the Constant Multiple Rule.

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Step 3.1

Multiply $\frac{e - x}{e}$ by $1$ .

$\frac{e - x}{e} \frac{d}{d x} [\frac{e}{e - x}]$

Step 3.2

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

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

Step 3.3

Simplify terms.

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Step 3.3.1

Combine $e$ and $\frac{e - x}{e}$ .

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

Step 3.3.2

Cancel the common factor of $e$ .

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Step 3.3.2.1

Cancel the common factor.

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

Step 3.3.2.2

Divide $e - x$ by $1$ .

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

Step 3.3.3

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

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

Step 4

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

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Step 4.1

To apply the Chain Rule, set $u_{2}$ as $e - x$ .

$(e - x) (\frac{d}{d u_{2}} [{u_{2}}^{- 1}] \frac{d}{d x} [e - x])$

Step 4.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = - 1$ .

$(e - x) (- {u_{2}}^{- 2} \frac{d}{d x} [e - x])$

Step 4.3

Replace all occurrences of $u_{2}$ with $e - x$ .

$(e - x) (- {(e - x)}^{- 2} \frac{d}{d x} [e - x])$

Step 5

Raise $e - x$ to the power of $1$ .

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

Step 6

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

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

Step 7

Subtract $2$ from $1$ .

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

Step 8

By the Sum Rule, the derivative of $e - x$ with respect to $x$ is $\frac{d}{d x} [e] + \frac{d}{d x} [- x]$ .

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

Step 9

Since $e$ is constant with respect to $x$ , the derivative of $e$ with respect to $x$ is $0$ .

$- {(e - x)}^{- 1} (0 + \frac{d}{d x} [- x])$

Step 10

Add $0$ and $\frac{d}{d x} [- x]$ .

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

Step 11

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

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

Step 12

Multiply.

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Step 12.1

Multiply $- 1$ by $- 1$ .

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

Step 12.2

Multiply ${(e - x)}^{- 1}$ by $1$ .

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

Step 13

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

${(e - x)}^{- 1} \cdot 1$

Step 14

Multiply ${(e - x)}^{- 1}$ by $1$ .

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

Step 15

Simplify.

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Step 15.1

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

$\frac{1}{e - x}$

Step 15.2

Reorder terms.

$\frac{1}{- x + e}$

Step 15.3

Factor $- 1$ out of $- x$ .

$\frac{1}{- (x) + e}$

Step 15.4

Factor $- 1$ out of $e$ .

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

Step 15.5

Factor $- 1$ out of $- (x) - 1 (- e)$ .

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

Step 15.6

Rewrite $- (x - e)$ as $- 1 (x - e)$ .

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

Step 15.7

Move the negative in front of the fraction.

$- \frac{1}{x - e}$