Calculus Examples

$f (x) = 5 \arctan (x^{2})$

Step 1

Find the first derivative.

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Step 1.1

Since $5$ is constant with respect to $x$ , the derivative of $5 \arctan (x^{2})$ with respect to $x$ is $5 \frac{d}{d x} [\arctan (x^{2})]$ .

$5 \frac{d}{d x} [\arctan (x^{2})]$

Step 1.2

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 $g (x) = x^{2}$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $x^{2}$ .

$5 (\frac{d}{d u} [\arctan (u)] \frac{d}{d x} [x^{2}])$

Step 1.2.2

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

$5 (\frac{1}{1 + u^{2}} \frac{d}{d x} [x^{2}])$

Step 1.2.3

Replace all occurrences of $u$ with $x^{2}$ .

$5 (\frac{1}{1 + {(x^{2})}^{2}} \frac{d}{d x} [x^{2}])$

Step 1.3

Differentiate using the Power Rule.

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Step 1.3.1

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 1.3.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$5 (\frac{1}{1 + x^{2 \cdot 2}} \frac{d}{d x} [x^{2}])$

Step 1.3.1.2

Multiply $2$ by $2$ .

$5 (\frac{1}{1 + x^{4}} \frac{d}{d x} [x^{2}])$

Step 1.3.2

Combine $\frac{1}{1 + x^{4}}$ and $5$ .

$\frac{5}{1 + x^{4}} \frac{d}{d x} [x^{2}]$

Step 1.3.3

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

$\frac{5}{1 + x^{4}} (2 x)$

Step 1.3.4

Combine fractions.

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Step 1.3.4.1

Combine $2$ and $\frac{5}{1 + x^{4}}$ .

$\frac{2 \cdot 5}{1 + x^{4}} x$

Step 1.3.4.2

Multiply $2$ by $5$ .

$\frac{10}{1 + x^{4}} x$

Step 1.3.4.3

Combine $\frac{10}{1 + x^{4}}$ and $x$ .

$\frac{10 x}{1 + x^{4}}$

Step 1.3.4.4

Reorder terms.

$f' (x) = \frac{10 x}{x^{4} + 1}$

Step 2

Find the second derivative.

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Step 2.1

Since $10$ is constant with respect to $x$ , the derivative of $\frac{10 x}{x^{4} + 1}$ with respect to $x$ is $10 \frac{d}{d x} [\frac{x}{x^{4} + 1}]$ .

$10 \frac{d}{d x} [\frac{x}{x^{4} + 1}]$

Step 2.2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x$ and $g (x) = x^{4} + 1$ .

$10 \frac{(x^{4} + 1) \frac{d}{d x} [x] - x \frac{d}{d x} [x^{4} + 1]}{{(x^{4} + 1)}^{2}}$

Step 2.3

Differentiate.

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Step 2.3.1

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

$10 \frac{(x^{4} + 1) \cdot 1 - x \frac{d}{d x} [x^{4} + 1]}{{(x^{4} + 1)}^{2}}$

Step 2.3.2

Multiply $x^{4} + 1$ by $1$ .

$10 \frac{x^{4} + 1 - x \frac{d}{d x} [x^{4} + 1]}{{(x^{4} + 1)}^{2}}$

Step 2.3.3

By the Sum Rule, the derivative of $x^{4} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [1]$ .

$10 \frac{x^{4} + 1 - x (\frac{d}{d x} [x^{4}] + \frac{d}{d x} [1])}{{(x^{4} + 1)}^{2}}$

Step 2.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$10 \frac{x^{4} + 1 - x (4 x^{3} + \frac{d}{d x} [1])}{{(x^{4} + 1)}^{2}}$

Step 2.3.5

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

$10 \frac{x^{4} + 1 - x (4 x^{3} + 0)}{{(x^{4} + 1)}^{2}}$

Step 2.3.6

Simplify the expression.

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Step 2.3.6.1

Add $4 x^{3}$ and $0$ .

$10 \frac{x^{4} + 1 - x (4 x^{3})}{{(x^{4} + 1)}^{2}}$

Step 2.3.6.2

Multiply $4$ by $- 1$ .

$10 \frac{x^{4} + 1 - 4 x \cdot x^{3}}{{(x^{4} + 1)}^{2}}$

Step 2.4

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 2.4.1

Move $x^{3}$ .

$10 \frac{x^{4} + 1 - 4 (x^{3} x)}{{(x^{4} + 1)}^{2}}$

Step 2.4.2

Multiply $x^{3}$ by $x$ .

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Step 2.4.2.1

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

$10 \frac{x^{4} + 1 - 4 (x^{3} x^{1})}{{(x^{4} + 1)}^{2}}$

Step 2.4.2.2

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

$10 \frac{x^{4} + 1 - 4 x^{3 + 1}}{{(x^{4} + 1)}^{2}}$

Step 2.4.3

Add $3$ and $1$ .

$10 \frac{x^{4} + 1 - 4 x^{4}}{{(x^{4} + 1)}^{2}}$

Step 2.5

Subtract $4 x^{4}$ from $x^{4}$ .

$10 \frac{- 3 x^{4} + 1}{{(x^{4} + 1)}^{2}}$

Step 2.6

Combine $10$ and $\frac{- 3 x^{4} + 1}{{(x^{4} + 1)}^{2}}$ .

$\frac{10 (- 3 x^{4} + 1)}{{(x^{4} + 1)}^{2}}$

Step 2.7

Simplify.

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Step 2.7.1

Apply the distributive property.

$\frac{10 (- 3 x^{4}) + 10 \cdot 1}{{(x^{4} + 1)}^{2}}$

Step 2.7.2

Simplify each term.

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Step 2.7.2.1

Multiply $- 3$ by $10$ .

$\frac{- 30 x^{4} + 10 \cdot 1}{{(x^{4} + 1)}^{2}}$

Step 2.7.2.2

Multiply $10$ by $1$ .

$\frac{- 30 x^{4} + 10}{{(x^{4} + 1)}^{2}}$

Step 2.7.3

Factor $10$ out of $- 30 x^{4} + 10$ .

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Step 2.7.3.1

Factor $10$ out of $- 30 x^{4}$ .

$\frac{10 (- 3 x^{4}) + 10}{{(x^{4} + 1)}^{2}}$

Step 2.7.3.2

Factor $10$ out of $10$ .

$\frac{10 (- 3 x^{4}) + 10 (1)}{{(x^{4} + 1)}^{2}}$

Step 2.7.3.3

Factor $10$ out of $10 (- 3 x^{4}) + 10 (1)$ .

$\frac{10 (- 3 x^{4} + 1)}{{(x^{4} + 1)}^{2}}$

Step 2.7.4

Factor $- 1$ out of $- 3 x^{4}$ .

$\frac{10 (- (3 x^{4}) + 1)}{{(x^{4} + 1)}^{2}}$

Step 2.7.5

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

$\frac{10 (- (3 x^{4}) - 1 (- 1))}{{(x^{4} + 1)}^{2}}$

Step 2.7.6

Factor $- 1$ out of $- (3 x^{4}) - 1 (- 1)$ .

$\frac{10 (- (3 x^{4} - 1))}{{(x^{4} + 1)}^{2}}$

Step 2.7.7

Rewrite $- (3 x^{4} - 1)$ as $- 1 (3 x^{4} - 1)$ .

$\frac{10 (- 1 (3 x^{4} - 1))}{{(x^{4} + 1)}^{2}}$

Step 2.7.8

Move the negative in front of the fraction.

$f'' (x) = - \frac{10 (3 x^{4} - 1)}{{(x^{4} + 1)}^{2}}$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $- \frac{10 (3 x^{4} - 1)}{{(x^{4} + 1)}^{2}}$ .

$- \frac{10 (3 x^{4} - 1)}{{(x^{4} + 1)}^{2}}$