Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Differentiate.

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

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

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

Step 3.3.5

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

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

Step 3.3.6

Simplify the expression.

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Step 3.3.6.1

Add $2 x$ and $0$ .

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

Step 3.3.6.2

Multiply $2$ by $- 1$ .

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

Add $1$ and $1$ .

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

Simplify.

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Step 3.10.1

Apply the distributive property.

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

Step 3.10.2

Simplify each term.

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Step 3.10.2.1

Multiply $- 1$ by $2$ .

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

Step 3.10.2.2

Multiply $2$ by $1$ .

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

Step 4

Reform the equation by setting the left side equal to the right side.

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

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

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