Calculus Examples

$\frac{n r}{1 + (n - 1) r}$

Step 1

Since $r$ is constant with respect to $n$ , the derivative of $\frac{n r}{1 + (n - 1) r}$ with respect to $n$ is $r \frac{d}{d n} [\frac{n}{1 + (n - 1) r}]$ .

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

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d n} [\frac{f (n)}{g (n)}]$ is $\frac{g (n) \frac{d}{d n} [f (n)] - f (n) \frac{d}{d n} [g (n)]}{{g (n)}^{2}}$ where $f (n) = n$ and $g (n) = 1 + (n - 1) r$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

Multiply $1 + (n - 1) r$ by $1$ .

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

Step 3.3

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

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

Step 3.4

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

$r \frac{1 + (n - 1) r - n (0 + \frac{d}{d n} [(n - 1) r])}{{(1 + (n - 1) r)}^{2}}$

Step 3.5

Add $0$ and $\frac{d}{d n} [(n - 1) r]$ .

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

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

$r \frac{1 + (n - 1) r - n r (1 + 0)}{{(1 + (n - 1) r)}^{2}}$

Step 3.10

Combine fractions.

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Step 3.10.1

Add $1$ and $0$ .

$r \frac{1 + (n - 1) r - n r \cdot 1}{{(1 + (n - 1) r)}^{2}}$

Step 3.10.2

Multiply $- 1$ by $1$ .

$r \frac{1 + (n - 1) r - n r}{{(1 + (n - 1) r)}^{2}}$

Step 3.10.3

Combine $r$ and $\frac{1 + (n - 1) r - n r}{{(1 + (n - 1) r)}^{2}}$ .

$\frac{r (1 + (n - 1) r - n r)}{{(1 + (n - 1) r)}^{2}}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\frac{r (1 + n r - 1 r - n r)}{{(1 + (n - 1) r)}^{2}}$

Step 4.2

Apply the distributive property.

$\frac{r \cdot 1 + r (n r) + r (- 1 r) + r (- n r)}{{(1 + (n - 1) r)}^{2}}$

Step 4.3

Apply the distributive property.

$\frac{r \cdot 1 + r (n r) + r (- 1 r) + r (- n r)}{{(1 + n r - 1 r)}^{2}}$

Step 4.4

Simplify the numerator.

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Step 4.4.1

Combine the opposite terms in $r \cdot 1 + r (n r) + r (- 1 r) + r (- n r)$ .

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Step 4.4.1.1

Reorder the factors in the terms $r (n r)$ and $r (- n r)$ .

$\frac{r \cdot 1 + n r \cdot r + r (- 1 r) - n r \cdot r}{{(1 + n r - 1 r)}^{2}}$

Step 4.4.1.2

Subtract $n r \cdot r$ from $n r \cdot r$ .

$\frac{r \cdot 1 + r (- 1 r) + 0}{{(1 + n r - 1 r)}^{2}}$

Step 4.4.1.3

Add $r \cdot 1 + r (- 1 r)$ and $0$ .

$\frac{r \cdot 1 + r (- 1 r)}{{(1 + n r - 1 r)}^{2}}$

Step 4.4.2

Simplify each term.

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Step 4.4.2.1

Multiply $r$ by $1$ .

$\frac{r + r (- 1 r)}{{(1 + n r - 1 r)}^{2}}$

Step 4.4.2.2

Rewrite using the commutative property of multiplication.

$\frac{r - r \cdot r}{{(1 + n r - 1 r)}^{2}}$

Step 4.4.2.3

Multiply $r$ by $r$ by adding the exponents.

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Step 4.4.2.3.1

Move $r$ .

$\frac{r - (r \cdot r)}{{(1 + n r - 1 r)}^{2}}$

Step 4.4.2.3.2

Multiply $r$ by $r$ .

$\frac{r - r^{2}}{{(1 + n r - 1 r)}^{2}}$

Step 4.5

Rewrite $- 1 r$ as $- r$ .

$\frac{r - r^{2}}{{(1 + n r - r)}^{2}}$

Step 4.6

Reorder terms.

$\frac{- r^{2} + r}{{(n r - r + 1)}^{2}}$

Step 4.7

Factor $r$ out of $- r^{2} + r$ .

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Step 4.7.1

Factor $r$ out of $- r^{2}$ .

$\frac{r (- r) + r}{{(n r - r + 1)}^{2}}$

Step 4.7.2

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

$\frac{r (- r) + r^{1}}{{(n r - r + 1)}^{2}}$

Step 4.7.3

Factor $r$ out of $r^{1}$ .

$\frac{r (- r) + r \cdot 1}{{(n r - r + 1)}^{2}}$

Step 4.7.4

Factor $r$ out of $r (- r) + r \cdot 1$ .

$\frac{r (- r + 1)}{{(n r - r + 1)}^{2}}$

Step 4.8

Factor $- 1$ out of $- r$ .

$\frac{r (- (r) + 1)}{{(n r - r + 1)}^{2}}$

Step 4.9

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

$\frac{r (- (r) - 1 (- 1))}{{(n r - r + 1)}^{2}}$

Step 4.10

Factor $- 1$ out of $- (r) - 1 (- 1)$ .

$\frac{r (- (r - 1))}{{(n r - r + 1)}^{2}}$

Step 4.11

Rewrite $- (r - 1)$ as $- 1 (r - 1)$ .

$\frac{r (- 1 (r - 1))}{{(n r - r + 1)}^{2}}$

Step 4.12

Move the negative in front of the fraction.

$- \frac{r (r - 1)}{{(n r - r + 1)}^{2}}$