Calculus Examples

$f (x) = n \ln (1 + \frac{x}{n})$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d n} [f (n) g (n)]$ is $f (n) \frac{d}{d n} [g (n)] + g (n) \frac{d}{d n} [f (n)]$ where $f (n) = n$ and $g (n) = \ln (1 + \frac{x}{n})$ .

$n \frac{d}{d n} [\ln (1 + \frac{x}{n})] + \ln (1 + \frac{x}{n}) \frac{d}{d n} [n]$

Step 2

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

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Step 2.1

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

$n (\frac{d}{d u} [\ln (u)] \frac{d}{d n} [1 + \frac{x}{n}]) + \ln (1 + \frac{x}{n}) \frac{d}{d n} [n]$

Step 2.2

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

$n (\frac{1}{u} \frac{d}{d n} [1 + \frac{x}{n}]) + \ln (1 + \frac{x}{n}) \frac{d}{d n} [n]$

Step 2.3

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

$n (\frac{1}{1 + \frac{x}{n}} \frac{d}{d n} [1 + \frac{x}{n}]) + \ln (1 + \frac{x}{n}) \frac{d}{d n} [n]$

Step 3

Differentiate.

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Step 3.1

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

$\frac{n}{1 + \frac{x}{n}} \frac{d}{d n} [1 + \frac{x}{n}] + \ln (1 + \frac{x}{n}) \frac{d}{d n} [n]$

Step 3.2

By the Sum Rule, the derivative of $1 + \frac{x}{n}$ with respect to $n$ is $\frac{d}{d n} [1] + \frac{d}{d n} [\frac{x}{n}]$ .

$\frac{n}{1 + \frac{x}{n}} (\frac{d}{d n} [1] + \frac{d}{d n} [\frac{x}{n}]) + \ln (1 + \frac{x}{n}) \frac{d}{d n} [n]$

Step 3.3

Since $1$ is constant with respect to $n$ , the derivative of $1$ with respect to $n$ is $0$ .

$\frac{n}{1 + \frac{x}{n}} (0 + \frac{d}{d n} [\frac{x}{n}]) + \ln (1 + \frac{x}{n}) \frac{d}{d n} [n]$

Step 3.4

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

$\frac{n}{1 + \frac{x}{n}} \frac{d}{d n} [\frac{x}{n}] + \ln (1 + \frac{x}{n}) \frac{d}{d n} [n]$

Step 3.5

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

$\frac{n}{1 + \frac{x}{n}} (x \frac{d}{d n} [\frac{1}{n}]) + \ln (1 + \frac{x}{n}) \frac{d}{d n} [n]$

Step 3.6

Combine fractions.

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Step 3.6.1

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

$\frac{x n}{1 + \frac{x}{n}} \frac{d}{d n} [\frac{1}{n}] + \ln (1 + \frac{x}{n}) \frac{d}{d n} [n]$

Step 3.6.2

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

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

Step 3.7

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

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

Step 3.8

Combine $\frac{x n}{1 + \frac{x}{n}}$ and $n^{- 2}$ .

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

Step 4

Multiply $n$ by $n^{- 2}$ by adding the exponents.

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Step 4.1

Move $n^{- 2}$ .

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

Step 4.2

Multiply $n^{- 2}$ by $n$ .

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.3

Add $- 2$ and $1$ .

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

Step 5

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

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

Step 6

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

$- \frac{x}{(1 + \frac{x}{n}) n} + \ln (1 + \frac{x}{n}) \cdot 1$

Step 7

Multiply $\ln (1 + \frac{x}{n})$ by $1$ .

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

Step 8

To write $\ln (1 + \frac{x}{n})$ as a fraction with a common denominator, multiply by $\frac{(1 + \frac{x}{n}) n}{(1 + \frac{x}{n}) n}$ .

$- \frac{x}{(1 + \frac{x}{n}) n} + \ln (1 + \frac{x}{n}) \cdot \frac{(1 + \frac{x}{n}) n}{(1 + \frac{x}{n}) n}$

Step 9

Combine $\ln (1 + \frac{x}{n})$ and $\frac{(1 + \frac{x}{n}) n}{(1 + \frac{x}{n}) n}$ .

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

Step 10

Combine the numerators over the common denominator.

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

Step 11

Simplify.

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Step 11.1

Apply the distributive property.

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

Step 11.2

Apply the distributive property.

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

Step 11.3

Simplify the numerator.

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Step 11.3.1

Simplify each term.

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Step 11.3.1.1

Simplify each term.

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Step 11.3.1.1.1

Multiply $n$ by $1$ .

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

Step 11.3.1.1.2

Cancel the common factor of $n$ .

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Step 11.3.1.1.2.1

Cancel the common factor.

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

Step 11.3.1.1.2.2

Rewrite the expression.

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

Step 11.3.1.2

Apply the distributive property.

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

Step 11.3.2

Reorder factors in $- x + \ln (1 + \frac{x}{n}) n + \ln (1 + \frac{x}{n}) x$ .

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

Step 11.4

Combine terms.

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Step 11.4.1

Multiply $n$ by $1$ .

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

Step 11.4.2

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

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

Step 11.4.3

Cancel the common factor of $n$ .

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Step 11.4.3.1

Cancel the common factor.

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

Step 11.4.3.2

Divide $x$ by $1$ .

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

Step 11.5

Reorder terms.

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

Step 11.6

Simplify each term.

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Step 11.6.1

Write $1$ as a fraction with a common denominator.

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

Step 11.6.2

Combine the numerators over the common denominator.

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

Step 11.6.3

Write $1$ as a fraction with a common denominator.

$\frac{n \ln (\frac{n + x}{n}) + x \ln (\frac{n}{n} + \frac{x}{n}) - x}{n + x}$

Step 11.6.4

Combine the numerators over the common denominator.

$\frac{n \ln (\frac{n + x}{n}) + x \ln (\frac{n + x}{n}) - x}{n + x}$