Calculus Examples

$f (x) = \arctan (\frac{30}{x}) - \arctan (\frac{10}{x})$

Step 1

By the Sum Rule, the derivative of $\arctan (\frac{30}{x}) - \arctan (\frac{10}{x})$ with respect to $x$ is $\frac{d}{d x} [\arctan (\frac{30}{x})] + \frac{d}{d x} [- \arctan (\frac{10}{x})]$ .

$\frac{d}{d x} [\arctan (\frac{30}{x})] + \frac{d}{d x} [- \arctan (\frac{10}{x})]$

Step 2

Evaluate $\frac{d}{d x} [\arctan (\frac{30}{x})]$ .

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Step 2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \arctan (x)$ and $g (x) = \frac{30}{x}$ .

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Step 2.1.1

To apply the Chain Rule, set $u_{1}$ as $\frac{30}{x}$ .

$\frac{d}{d u_{1}} [\arctan (u_{1})] \frac{d}{d x} [\frac{30}{x}] + \frac{d}{d x} [- \arctan (\frac{10}{x})]$

Step 2.1.2

The derivative of $\arctan (u_{1})$ with respect to $u_{1}$ is $\frac{1}{1 + {u_{1}}^{2}}$ .

$\frac{1}{1 + {u_{1}}^{2}} \frac{d}{d x} [\frac{30}{x}] + \frac{d}{d x} [- \arctan (\frac{10}{x})]$

Step 2.1.3

Replace all occurrences of $u_{1}$ with $\frac{30}{x}$ .

$\frac{1}{1 + {(\frac{30}{x})}^{2}} \frac{d}{d x} [\frac{30}{x}] + \frac{d}{d x} [- \arctan (\frac{10}{x})]$

Step 2.2

Since $30$ is constant with respect to $x$ , the derivative of $\frac{30}{x}$ with respect to $x$ is $30 \frac{d}{d x} [\frac{1}{x}]$ .

$\frac{1}{1 + {(\frac{30}{x})}^{2}} (30 \frac{d}{d x} [\frac{1}{x}]) + \frac{d}{d x} [- \arctan (\frac{10}{x})]$

Step 2.3

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

$\frac{1}{1 + {(\frac{30}{x})}^{2}} (30 \frac{d}{d x} [x^{- 1}]) + \frac{d}{d x} [- \arctan (\frac{10}{x})]$

Step 2.4

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

$\frac{1}{1 + {(\frac{30}{x})}^{2}} (30 (- x^{- 2})) + \frac{d}{d x} [- \arctan (\frac{10}{x})]$

Step 2.5

Multiply $- 1$ by $30$ .

$\frac{1}{1 + {(\frac{30}{x})}^{2}} (- 30 x^{- 2}) + \frac{d}{d x} [- \arctan (\frac{10}{x})]$

Step 2.6

Combine $- 30$ and $\frac{1}{1 + {(\frac{30}{x})}^{2}}$ .

$\frac{- 30}{1 + {(\frac{30}{x})}^{2}} x^{- 2} + \frac{d}{d x} [- \arctan (\frac{10}{x})]$

Step 2.7

Combine $\frac{- 30}{1 + {(\frac{30}{x})}^{2}}$ and $x^{- 2}$ .

$\frac{- 30 x^{- 2}}{1 + {(\frac{30}{x})}^{2}} + \frac{d}{d x} [- \arctan (\frac{10}{x})]$

Step 2.8

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{- 30}{(1 + {(\frac{30}{x})}^{2}) x^{2}} + \frac{d}{d x} [- \arctan (\frac{10}{x})]$

Step 2.9

Move the negative in front of the fraction.

$- \frac{30}{(1 + {(\frac{30}{x})}^{2}) x^{2}} + \frac{d}{d x} [- \arctan (\frac{10}{x})]$

Step 3

Evaluate $\frac{d}{d x} [- \arctan (\frac{10}{x})]$ .

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Step 3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \arctan (\frac{10}{x})$ with respect to $x$ is $- \frac{d}{d x} [\arctan (\frac{10}{x})]$ .

$- \frac{30}{(1 + {(\frac{30}{x})}^{2}) x^{2}} - \frac{d}{d x} [\arctan (\frac{10}{x})]$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \arctan (x)$ and $g (x) = \frac{10}{x}$ .

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Step 3.2.1

To apply the Chain Rule, set $u_{2}$ as $\frac{10}{x}$ .

$- \frac{30}{(1 + {(\frac{30}{x})}^{2}) x^{2}} - (\frac{d}{d u_{2}} [\arctan (u_{2})] \frac{d}{d x} [\frac{10}{x}])$

Step 3.2.2

The derivative of $\arctan (u_{2})$ with respect to $u_{2}$ is $\frac{1}{1 + {u_{2}}^{2}}$ .

$- \frac{30}{(1 + {(\frac{30}{x})}^{2}) x^{2}} - (\frac{1}{1 + {u_{2}}^{2}} \frac{d}{d x} [\frac{10}{x}])$

Step 3.2.3

Replace all occurrences of $u_{2}$ with $\frac{10}{x}$ .

$- \frac{30}{(1 + {(\frac{30}{x})}^{2}) x^{2}} - (\frac{1}{1 + {(\frac{10}{x})}^{2}} \frac{d}{d x} [\frac{10}{x}])$

Step 3.3

Since $10$ is constant with respect to $x$ , the derivative of $\frac{10}{x}$ with respect to $x$ is $10 \frac{d}{d x} [\frac{1}{x}]$ .

$- \frac{30}{(1 + {(\frac{30}{x})}^{2}) x^{2}} - (\frac{1}{1 + {(\frac{10}{x})}^{2}} (10 \frac{d}{d x} [\frac{1}{x}]))$

Step 3.4

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

$- \frac{30}{(1 + {(\frac{30}{x})}^{2}) x^{2}} - (\frac{1}{1 + {(\frac{10}{x})}^{2}} (10 \frac{d}{d x} [x^{- 1}]))$

Step 3.5

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

$- \frac{30}{(1 + {(\frac{30}{x})}^{2}) x^{2}} - (\frac{1}{1 + {(\frac{10}{x})}^{2}} (10 (- x^{- 2})))$

Step 3.6

Multiply $- 1$ by $10$ .

$- \frac{30}{(1 + {(\frac{30}{x})}^{2}) x^{2}} - (\frac{1}{1 + {(\frac{10}{x})}^{2}} (- 10 x^{- 2}))$

Step 3.7

Combine $- 10$ and $\frac{1}{1 + {(\frac{10}{x})}^{2}}$ .

$- \frac{30}{(1 + {(\frac{30}{x})}^{2}) x^{2}} - (\frac{- 10}{1 + {(\frac{10}{x})}^{2}} x^{- 2})$

Step 3.8

Combine $\frac{- 10}{1 + {(\frac{10}{x})}^{2}}$ and $x^{- 2}$ .

$- \frac{30}{(1 + {(\frac{30}{x})}^{2}) x^{2}} - \frac{- 10 x^{- 2}}{1 + {(\frac{10}{x})}^{2}}$

Step 3.9

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{30}{(1 + {(\frac{30}{x})}^{2}) x^{2}} - \frac{- 10}{(1 + {(\frac{10}{x})}^{2}) x^{2}}$

Step 3.10

Move the negative in front of the fraction.

$- \frac{30}{(1 + {(\frac{30}{x})}^{2}) x^{2}} - - \frac{10}{(1 + {(\frac{10}{x})}^{2}) x^{2}}$

Step 3.11

Multiply $- 1$ by $- 1$ .

$- \frac{30}{(1 + {(\frac{30}{x})}^{2}) x^{2}} + 1 \frac{10}{(1 + {(\frac{10}{x})}^{2}) x^{2}}$

Step 3.12

Multiply $\frac{10}{(1 + {(\frac{10}{x})}^{2}) x^{2}}$ by $1$ .

$- \frac{30}{(1 + {(\frac{30}{x})}^{2}) x^{2}} + \frac{10}{(1 + {(\frac{10}{x})}^{2}) x^{2}}$

Step 4

Simplify.

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Step 4.1

Apply the product rule to $\frac{30}{x}$ .

$- \frac{30}{(1 + \frac{30^{2}}{x^{2}}) x^{2}} + \frac{10}{(1 + {(\frac{10}{x})}^{2}) x^{2}}$

Step 4.2

Apply the product rule to $\frac{10}{x}$ .

$- \frac{30}{(1 + \frac{30^{2}}{x^{2}}) x^{2}} + \frac{10}{(1 + \frac{10^{2}}{x^{2}}) x^{2}}$

Step 4.3

Apply the distributive property.

$- \frac{30}{1 x^{2} + \frac{30^{2}}{x^{2}} x^{2}} + \frac{10}{(1 + \frac{10^{2}}{x^{2}}) x^{2}}$

Step 4.4

Apply the distributive property.

$- \frac{30}{1 x^{2} + \frac{30^{2}}{x^{2}} x^{2}} + \frac{10}{1 x^{2} + \frac{10^{2}}{x^{2}} x^{2}}$

Step 4.5

Combine terms.

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Step 4.5.1

Multiply $x^{2}$ by $1$ .

$- \frac{30}{x^{2} + \frac{30^{2}}{x^{2}} x^{2}} + \frac{10}{1 x^{2} + \frac{10^{2}}{x^{2}} x^{2}}$

Step 4.5.2

Raise $30$ to the power of $2$ .

$- \frac{30}{x^{2} + \frac{900}{x^{2}} x^{2}} + \frac{10}{1 x^{2} + \frac{10^{2}}{x^{2}} x^{2}}$

Step 4.5.3

Combine $\frac{900}{x^{2}}$ and $x^{2}$ .

$- \frac{30}{x^{2} + \frac{900 x^{2}}{x^{2}}} + \frac{10}{1 x^{2} + \frac{10^{2}}{x^{2}} x^{2}}$

Step 4.5.4

Cancel the common factor of $x^{2}$ .

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Step 4.5.4.1

Cancel the common factor.

$- \frac{30}{x^{2} + \frac{900 x^{2}}{x^{2}}} + \frac{10}{1 x^{2} + \frac{10^{2}}{x^{2}} x^{2}}$

Step 4.5.4.2

Divide $900$ by $1$ .

$- \frac{30}{x^{2} + 900} + \frac{10}{1 x^{2} + \frac{10^{2}}{x^{2}} x^{2}}$

Step 4.5.5

Multiply $x^{2}$ by $1$ .

$- \frac{30}{x^{2} + 900} + \frac{10}{x^{2} + \frac{10^{2}}{x^{2}} x^{2}}$

Step 4.5.6

Raise $10$ to the power of $2$ .

$- \frac{30}{x^{2} + 900} + \frac{10}{x^{2} + \frac{100}{x^{2}} x^{2}}$

Step 4.5.7

Combine $\frac{100}{x^{2}}$ and $x^{2}$ .

$- \frac{30}{x^{2} + 900} + \frac{10}{x^{2} + \frac{100 x^{2}}{x^{2}}}$

Step 4.5.8

Cancel the common factor of $x^{2}$ .

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Step 4.5.8.1

Cancel the common factor.

$- \frac{30}{x^{2} + 900} + \frac{10}{x^{2} + \frac{100 x^{2}}{x^{2}}}$

Step 4.5.8.2

Divide $100$ by $1$ .

$- \frac{30}{x^{2} + 900} + \frac{10}{x^{2} + 100}$

Step 4.5.9

To write $- \frac{30}{x^{2} + 900}$ as a fraction with a common denominator, multiply by $\frac{x^{2} + 100}{x^{2} + 100}$ .

$- \frac{30}{x^{2} + 900} \cdot \frac{x^{2} + 100}{x^{2} + 100} + \frac{10}{x^{2} + 100}$

Step 4.5.10

To write $\frac{10}{x^{2} + 100}$ as a fraction with a common denominator, multiply by $\frac{x^{2} + 900}{x^{2} + 900}$ .

$- \frac{30}{x^{2} + 900} \cdot \frac{x^{2} + 100}{x^{2} + 100} + \frac{10}{x^{2} + 100} \cdot \frac{x^{2} + 900}{x^{2} + 900}$

Step 4.5.11

Write each expression with a common denominator of $(x^{2} + 900) (x^{2} + 100)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.5.11.1

Multiply $\frac{30}{x^{2} + 900}$ by $\frac{x^{2} + 100}{x^{2} + 100}$ .

$- \frac{30 (x^{2} + 100)}{(x^{2} + 900) (x^{2} + 100)} + \frac{10}{x^{2} + 100} \cdot \frac{x^{2} + 900}{x^{2} + 900}$

Step 4.5.11.2

Multiply $\frac{10}{x^{2} + 100}$ by $\frac{x^{2} + 900}{x^{2} + 900}$ .

$- \frac{30 (x^{2} + 100)}{(x^{2} + 900) (x^{2} + 100)} + \frac{10 (x^{2} + 900)}{(x^{2} + 100) (x^{2} + 900)}$

Step 4.5.11.3

Reorder the factors of $(x^{2} + 900) (x^{2} + 100)$ .

$- \frac{30 (x^{2} + 100)}{(x^{2} + 100) (x^{2} + 900)} + \frac{10 (x^{2} + 900)}{(x^{2} + 100) (x^{2} + 900)}$

Step 4.5.12

Combine the numerators over the common denominator.

$\frac{- 30 (x^{2} + 100) + 10 (x^{2} + 900)}{(x^{2} + 100) (x^{2} + 900)}$

Step 4.6

Reorder terms.

$\frac{10 (x^{2} + 900) - 30 (x^{2} + 100)}{(x^{2} + 100) (x^{2} + 900)}$

Step 4.7

Simplify the numerator.

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Step 4.7.1

Factor $10$ out of $10 (x^{2} + 900) - 30 (x^{2} + 100)$ .

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Step 4.7.1.1

Factor $10$ out of $- 30 (x^{2} + 100)$ .

$\frac{10 (x^{2} + 900) + 10 (- 3 (x^{2} + 100))}{(x^{2} + 100) (x^{2} + 900)}$

Step 4.7.1.2

Factor $10$ out of $10 (x^{2} + 900) + 10 (- 3 (x^{2} + 100))$ .

$\frac{10 (x^{2} + 900 - 3 (x^{2} + 100))}{(x^{2} + 100) (x^{2} + 900)}$

Step 4.7.2

Apply the distributive property.

$\frac{10 (x^{2} + 900 - 3 x^{2} - 3 \cdot 100)}{(x^{2} + 100) (x^{2} + 900)}$

Step 4.7.3

Multiply $- 3$ by $100$ .

$\frac{10 (x^{2} + 900 - 3 x^{2} - 300)}{(x^{2} + 100) (x^{2} + 900)}$

Step 4.7.4

Subtract $3 x^{2}$ from $x^{2}$ .

$\frac{10 (- 2 x^{2} + 900 - 300)}{(x^{2} + 100) (x^{2} + 900)}$

Step 4.7.5

Subtract $300$ from $900$ .

$\frac{10 (- 2 x^{2} + 600)}{(x^{2} + 100) (x^{2} + 900)}$

Step 4.7.6

Factor $2$ out of $- 2 x^{2} + 600$ .

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Step 4.7.6.1

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

$\frac{10 (2 (- x^{2}) + 600)}{(x^{2} + 100) (x^{2} + 900)}$

Step 4.7.6.2

Factor $2$ out of $600$ .

$\frac{10 (2 (- x^{2}) + 2 (300))}{(x^{2} + 100) (x^{2} + 900)}$

Step 4.7.6.3

Factor $2$ out of $2 (- x^{2}) + 2 (300)$ .

$\frac{10 (2 (- x^{2} + 300))}{(x^{2} + 100) (x^{2} + 900)}$

$\frac{10 \cdot 2 (- x^{2} + 300)}{(x^{2} + 100) (x^{2} + 900)}$

Step 4.7.7

Multiply $10$ by $2$ .

$\frac{20 (- x^{2} + 300)}{(x^{2} + 100) (x^{2} + 900)}$

Step 4.8

Factor $- 1$ out of $- x^{2}$ .

$\frac{20 (- (x^{2}) + 300)}{(x^{2} + 100) (x^{2} + 900)}$

Step 4.9

Rewrite $300$ as $- 1 (- 300)$ .

$\frac{20 (- (x^{2}) - 1 (- 300))}{(x^{2} + 100) (x^{2} + 900)}$

Step 4.10

Factor $- 1$ out of $- (x^{2}) - 1 (- 300)$ .

$\frac{20 (- (x^{2} - 300))}{(x^{2} + 100) (x^{2} + 900)}$

Step 4.11

Rewrite $- (x^{2} - 300)$ as $- 1 (x^{2} - 300)$ .

$\frac{20 (- 1 (x^{2} - 300))}{(x^{2} + 100) (x^{2} + 900)}$

Step 4.12

Move the negative in front of the fraction.

$- \frac{20 (x^{2} - 300)}{(x^{2} + 100) (x^{2} + 900)}$