Calculus Examples

$A = 2000 {(1 + \frac{r}{12})}^{60}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d A} (A) = \frac{d}{d A} (2000 {(1 + \frac{r}{12})}^{60})$

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $2000$ is constant with respect to $A$ , the derivative of $2000 {(1 + \frac{r}{12})}^{60}$ with respect to $A$ is $2000 \frac{d}{d A} [{(1 + \frac{r}{12})}^{60}]$ .

$2000 \frac{d}{d A} [{(1 + \frac{r}{12})}^{60}]$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d A} [f (g (A))]$ is $f' (g (A)) g' (A)$ where $f (A) = A^{60}$ and $g (A) = 1 + \frac{r}{12}$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $1 + \frac{r}{12}$ .

$2000 (\frac{d}{d u} [u^{60}] \frac{d}{d A} [1 + \frac{r}{12}])$

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 60$ .

$2000 (60 u^{59} \frac{d}{d A} [1 + \frac{r}{12}])$

Step 3.2.3

Replace all occurrences of $u$ with $1 + \frac{r}{12}$ .

$2000 (60 {(1 + \frac{r}{12})}^{59} \frac{d}{d A} [1 + \frac{r}{12}])$

Step 3.3

Differentiate.

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Step 3.3.1

Multiply $60$ by $2000$ .

$120000 ({(1 + \frac{r}{12})}^{59} \frac{d}{d A} [1 + \frac{r}{12}])$

Step 3.3.2

By the Sum Rule, the derivative of $1 + \frac{r}{12}$ with respect to $A$ is $\frac{d}{d A} [1] + \frac{d}{d A} [\frac{r}{12}]$ .

$120000 {(1 + \frac{r}{12})}^{59} (\frac{d}{d A} [1] + \frac{d}{d A} [\frac{r}{12}])$

Step 3.3.3

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

$120000 {(1 + \frac{r}{12})}^{59} (0 + \frac{d}{d A} [\frac{r}{12}])$

Step 3.3.4

Add $0$ and $\frac{d}{d A} [\frac{r}{12}]$ .

$120000 {(1 + \frac{r}{12})}^{59} \frac{d}{d A} [\frac{r}{12}]$

Step 3.3.5

Since $\frac{1}{12}$ is constant with respect to $A$ , the derivative of $\frac{r}{12}$ with respect to $A$ is $\frac{1}{12} \frac{d}{d A} [r]$ .

$120000 {(1 + \frac{r}{12})}^{59} (\frac{1}{12} \frac{d}{d A} [r])$

Step 3.3.6

Simplify terms.

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Step 3.3.6.1

Combine $\frac{1}{12}$ and $120000$ .

$\frac{120000}{12} {(1 + \frac{r}{12})}^{59} \frac{d}{d A} [r]$

Step 3.3.6.2

Combine $\frac{120000}{12}$ and ${(1 + \frac{r}{12})}^{59}$ .

$\frac{120000 {(1 + \frac{r}{12})}^{59}}{12} \frac{d}{d A} [r]$

Step 3.3.6.3

Cancel the common factor of $120000$ and $12$ .

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Step 3.3.6.3.1

Factor $12$ out of $120000 {(1 + \frac{r}{12})}^{59}$ .

$\frac{12 (10000 {(1 + \frac{r}{12})}^{59})}{12} \frac{d}{d A} [r]$

Step 3.3.6.3.2

Cancel the common factors.

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Step 3.3.6.3.2.1

Factor $12$ out of $12$ .

$\frac{12 (10000 {(1 + \frac{r}{12})}^{59})}{12 (1)} \frac{d}{d A} [r]$

Step 3.3.6.3.2.2

Cancel the common factor.

$\frac{12 (10000 {(1 + \frac{r}{12})}^{59})}{12 \cdot 1} \frac{d}{d A} [r]$

Step 3.3.6.3.2.3

Rewrite the expression.

$\frac{10000 {(1 + \frac{r}{12})}^{59}}{1} \frac{d}{d A} [r]$

Step 3.3.6.3.2.4

Divide $10000 {(1 + \frac{r}{12})}^{59}$ by $1$ .

$10000 {(1 + \frac{r}{12})}^{59} \frac{d}{d A} [r]$

Step 3.4

Rewrite $\frac{d}{d A} [r]$ as $r'$ .

$10000 {(1 + \frac{r}{12})}^{59} r'$

Step 4

Reform the equation by setting the left side equal to the right side.

$1 = 10000 {(1 + \frac{r}{12})}^{59} r'$

Step 5

Solve for $r'$ .

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Step 5.1

Rewrite the equation as $10000 {(1 + \frac{r}{12})}^{59} r' = 1$ .

$10000 {(1 + \frac{r}{12})}^{59} r' = 1$

Step 5.2

Divide each term in $10000 {(1 + \frac{r}{12})}^{59} r' = 1$ by $10000 {(1 + \frac{r}{12})}^{59}$ and simplify.

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Step 5.2.1

Divide each term in $10000 {(1 + \frac{r}{12})}^{59} r' = 1$ by $10000 {(1 + \frac{r}{12})}^{59}$ .

$\frac{10000 {(1 + \frac{r}{12})}^{59} r'}{10000 {(1 + \frac{r}{12})}^{59}} = \frac{1}{10000 {(1 + \frac{r}{12})}^{59}}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $10000$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{10000 {(1 + \frac{r}{12})}^{59} r'}{10000 {(1 + \frac{r}{12})}^{59}} = \frac{1}{10000 {(1 + \frac{r}{12})}^{59}}$

Step 5.2.2.1.2

Rewrite the expression.

$\frac{{(1 + \frac{r}{12})}^{59} r'}{{(1 + \frac{r}{12})}^{59}} = \frac{1}{10000 {(1 + \frac{r}{12})}^{59}}$

Step 5.2.2.2

Cancel the common factor of ${(1 + \frac{r}{12})}^{59}$ .

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Step 5.2.2.2.1

Cancel the common factor.

$\frac{{(1 + \frac{r}{12})}^{59} r'}{{(1 + \frac{r}{12})}^{59}} = \frac{1}{10000 {(1 + \frac{r}{12})}^{59}}$

Step 5.2.2.2.2

Divide $r'$ by $1$ .

$r' = \frac{1}{10000 {(1 + \frac{r}{12})}^{59}}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Simplify the denominator.

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Step 5.2.3.1.1

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

$r' = \frac{1}{10000 {(\frac{12}{12} + \frac{r}{12})}^{59}}$

Step 5.2.3.1.2

Combine the numerators over the common denominator.

$r' = \frac{1}{10000 {(\frac{12 + r}{12})}^{59}}$

Step 5.2.3.1.3

Apply the product rule to $\frac{12 + r}{12}$ .

$r' = \frac{1}{10000 \frac{{(12 + r)}^{59}}{12^{59}}}$

Step 5.2.3.2

Combine $10000$ and $\frac{{(12 + r)}^{59}}{12^{59}}$ .

$r' = \frac{1}{\frac{10000 {(12 + r)}^{59}}{12^{59}}}$

Step 5.2.3.3

Multiply the numerator by the reciprocal of the denominator.

$r' = 1 \frac{12^{59}}{10000 {(12 + r)}^{59}}$

Step 5.2.3.4

Multiply $\frac{12^{59}}{10000 {(12 + r)}^{59}}$ by $1$ .

$r' = \frac{12^{59}}{10000 {(12 + r)}^{59}}$

Step 6

Replace $r'$ with $\frac{d r}{d A}$ .

$\frac{d r}{d A} = \frac{12^{59}}{10000 {(12 + r)}^{59}}$