Calculus Examples

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

Step 1

Find the first derivative.

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

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

Reorder terms.

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

Step 2

Find the second derivative.

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

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

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

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

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

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

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

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

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

Differentiate.

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

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

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

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

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

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

Simplify the expression.

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Add $2 x$ and $0$ .

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

Multiply $2$ by $- 1$ .

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

Simplify.

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

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

Combine terms.

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

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

Move the negative in front of the fraction.

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

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

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

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

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

Step 3

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

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