Calculus Examples

$\arctan (x)$

Step 1

Write $\arctan (x)$ as a function.

$f (x) = \arctan (x)$

Step 2

Find the $x$ values where the second derivative is equal to $0$ .

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Step 2.1

Find the second derivative.

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Step 2.1.1

Find the first derivative.

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Step 2.1.1.1

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

$\frac{1}{1 + x^{2}}$

Step 2.1.1.2

Reorder terms.

$f' (x) = \frac{1}{x^{2} + 1}$

Step 2.1.2

Find the second derivative.

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Step 2.1.2.1

Rewrite $\frac{1}{x^{2} + 1}$ as ${(x^{2} + 1)}^{- 1}$ .

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

Step 2.1.2.2

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

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Step 2.1.2.2.1

To apply the Chain Rule, set $u$ as $x^{2} + 1$ .

$\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2} + 1]$

Step 2.1.2.2.2

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

$- u^{- 2} \frac{d}{d x} [x^{2} + 1]$

Step 2.1.2.2.3

Replace all occurrences of $u$ with $x^{2} + 1$ .

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

Step 2.1.2.3

Differentiate.

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Step 2.1.2.3.1

By the Sum Rule, the derivative of $x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$ .

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

Step 2.1.2.3.2

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

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

Step 2.1.2.3.3

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

$- {(x^{2} + 1)}^{- 2} (2 x + 0)$

Step 2.1.2.3.4

Simplify the expression.

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Step 2.1.2.3.4.1

Add $2 x$ and $0$ .

$- {(x^{2} + 1)}^{- 2} (2 x)$

Step 2.1.2.3.4.2

Multiply $2$ by $- 1$ .

$- 2 {(x^{2} + 1)}^{- 2} x$

Step 2.1.2.4

Simplify.

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Step 2.1.2.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.1.2.4.2

Combine terms.

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Step 2.1.2.4.2.1

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

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

Step 2.1.2.4.2.2

Move the negative in front of the fraction.

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

Step 2.1.2.4.2.3

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

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

Step 2.1.2.4.2.4

Move $2$ to the left of $x$ .

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

Step 2.1.3

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

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

Step 2.2

Set the second derivative equal to $0$ then solve the equation $- \frac{2 x}{{(x^{2} + 1)}^{2}} = 0$ .

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Step 2.2.1

Set the second derivative equal to $0$ .

$- \frac{2 x}{{(x^{2} + 1)}^{2}} = 0$

Step 2.2.2

Set the numerator equal to zero.

$2 x = 0$

Step 2.2.3

Divide each term in $2 x = 0$ by $2$ and simplify.

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Step 2.2.3.1

Divide each term in $2 x = 0$ by $2$ .

$\frac{2 x}{2} = \frac{0}{2}$

Step 2.2.3.2

Simplify the left side.

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Step 2.2.3.2.1

Cancel the common factor of $2$ .

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Step 2.2.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{0}{2}$

Step 2.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{2}$

Step 2.2.3.3

Simplify the right side.

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Step 2.2.3.3.1

Divide $0$ by $2$ .

$x = 0$

Step 3

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4

Create intervals around the $x$ -values where the second derivative is zero or undefined.

$(- \infty, 0) \cup (0, \infty)$

Step 5

Substitute any number from the interval $(- \infty, 0)$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

Replace the variable $x$ with $- 2$ in the expression.

$f'' (- 2) = - \frac{2 (- 2)}{{({(- 2)}^{2} + 1)}^{2}}$

Step 5.2

Simplify the result.

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Step 5.2.1

Multiply $2$ by $- 2$ .

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

Step 5.2.2

Simplify the denominator.

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Step 5.2.2.1

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

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

Step 5.2.2.2

Add $4$ and $1$ .

$f'' (- 2) = - \frac{- 4}{5^{2}}$

Step 5.2.2.3

Raise $5$ to the power of $2$ .

$f'' (- 2) = - \frac{- 4}{25}$

Step 5.2.3

Move the negative in front of the fraction.

$f'' (- 2) = \frac{4}{25}$

Step 5.2.4

The final answer is $\frac{4}{25}$ .

$\frac{4}{25}$

Step 5.3

The graph is concave up on the interval $(- \infty, 0)$ because $f'' (- 2)$ is positive.

Concave up on $(- \infty, 0)$ since $f'' (x)$ is positive

Step 6

Substitute any number from the interval $(0, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

Replace the variable $x$ with $2$ in the expression.

$f'' (2) = - \frac{2 (2)}{{({(2)}^{2} + 1)}^{2}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Multiply $2$ by $2$ .

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

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

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

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

Step 6.2.2.2

Add $4$ and $1$ .

$f'' (2) = - \frac{4}{5^{2}}$

Step 6.2.2.3

Raise $5$ to the power of $2$ .

$f'' (2) = - \frac{4}{25}$

Step 6.2.3

The final answer is $- \frac{4}{25}$ .

$- \frac{4}{25}$

Step 6.3

The graph is concave down on the interval $(0, \infty)$ because $f'' (2)$ is negative.

Concave down on $(0, \infty)$ since $f'' (x)$ is negative

Step 7

The graph is concave down when the second derivative is negative and concave up when the second derivative is positive.

Concave up on $(- \infty, 0)$ since $f'' (x)$ is positive

Concave down on $(0, \infty)$ since $f'' (x)$ is negative

Step 8