Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

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

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

Step 1.1.2

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

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Step 1.1.2.1

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 1.1.2.2

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 1.1.2.3

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

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

Step 1.1.3

Differentiate.

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Step 1.1.3.1

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 1.1.3.2

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 1.1.3.3

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 1.1.3.4

Simplify the expression.

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Step 1.1.3.4.1

Add $2 x$ and $0$ .

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

Step 1.1.3.4.2

Multiply $2$ by $- 1$ .

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

Step 1.1.4

Simplify.

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Step 1.1.4.1

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

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

Step 1.1.4.2

Combine terms.

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Step 1.1.4.2.1

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

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

Step 1.1.4.2.2

Move the negative in front of the fraction.

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

Step 1.1.4.2.3

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

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

Step 1.1.4.2.4

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

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

Step 1.2

The first 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

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

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Set the numerator equal to zero.

$2 x = 0$

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $2$ .

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Step 2.3.2.1.1

Cancel the common factor.

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

Step 2.3.2.1.2

Divide $x$ by $1$ .

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

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $0$ by $2$ .

$x = 0$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

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.

Step 4

Evaluate $\frac{1}{x^{2} + 1}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 0$ .

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Step 4.1.1

Substitute $0$ for $x$ .

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

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify the denominator.

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Step 4.1.2.1.1

Raising $0$ to any positive power yields $0$ .

$\frac{1}{0 + 1}$

Step 4.1.2.1.2

Add $0$ and $1$ .

$\frac{1}{1}$

Step 4.1.2.2

Divide $1$ by $1$ .

$1$

Step 4.2

List all of the points.

$(0, 1)$

Step 5