Calculus Examples

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

Step 1

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 $g (x) = x^{2} + 1$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Add $2 x$ and $0$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Simplify the numerator.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

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

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

Step 3.2.1.2

Multiply $1$ by $1$ .

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

Step 3.2.1.3

Rewrite using the commutative property of multiplication.

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

Step 3.2.1.4

Multiply $x$ by $x$ by adding the exponents.

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Step 3.2.1.4.1

Move $x$ .

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

Step 3.2.1.4.2

Multiply $x$ by $x$ .

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

Step 3.2.1.5

Multiply $- 1$ by $2$ .

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

Step 3.2.2

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

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

Step 3.3

Simplify the numerator.

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Step 3.3.1

Rewrite $1$ as $1^{2}$ .

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

Step 3.3.2

Reorder $- x^{2}$ and $1^{2}$ .

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

Step 3.3.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = x$ .

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