Calculus Examples

$\frac{2 x}{1 + x^{4}}$

Step 1

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2 x}{1 + x^{4}}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{x}{1 + x^{4}}]$ .

$2 \frac{d}{d x} [\frac{x}{1 + x^{4}}]$

Step 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 $g (x) = 1 + x^{4}$ .

$2 \frac{(1 + x^{4}) \frac{d}{d x} [x] - x \frac{d}{d x} [1 + x^{4}]}{{(1 + x^{4})}^{2}}$

Step 3

Differentiate.

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Step 3.1

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

$2 \frac{(1 + x^{4}) \cdot 1 - x \frac{d}{d x} [1 + x^{4}]}{{(1 + x^{4})}^{2}}$

Step 3.2

Multiply $1 + x^{4}$ by $1$ .

$2 \frac{1 + x^{4} - x \frac{d}{d x} [1 + x^{4}]}{{(1 + x^{4})}^{2}}$

Step 3.3

By the Sum Rule, the derivative of $1 + x^{4}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [x^{4}]$ .

$2 \frac{1 + x^{4} - x (\frac{d}{d x} [1] + \frac{d}{d x} [x^{4}])}{{(1 + x^{4})}^{2}}$

Step 3.4

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

$2 \frac{1 + x^{4} - x (0 + \frac{d}{d x} [x^{4}])}{{(1 + x^{4})}^{2}}$

Step 3.5

Add $0$ and $\frac{d}{d x} [x^{4}]$ .

$2 \frac{1 + x^{4} - x \frac{d}{d x} [x^{4}]}{{(1 + x^{4})}^{2}}$

Step 3.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$2 \frac{1 + x^{4} - x (4 x^{3})}{{(1 + x^{4})}^{2}}$

Step 3.7

Multiply $4$ by $- 1$ .

$2 \frac{1 + x^{4} - 4 x \cdot x^{3}}{{(1 + x^{4})}^{2}}$

Step 4

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 4.1

Move $x^{3}$ .

$2 \frac{1 + x^{4} - 4 (x^{3} x)}{{(1 + x^{4})}^{2}}$

Step 4.2

Multiply $x^{3}$ by $x$ .

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Step 4.2.1

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

$2 \frac{1 + x^{4} - 4 (x^{3} x^{1})}{{(1 + x^{4})}^{2}}$

Step 4.2.2

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

$2 \frac{1 + x^{4} - 4 x^{3 + 1}}{{(1 + x^{4})}^{2}}$

Step 4.3

Add $3$ and $1$ .

$2 \frac{1 + x^{4} - 4 x^{4}}{{(1 + x^{4})}^{2}}$

Step 5

Subtract $4 x^{4}$ from $x^{4}$ .

$2 \frac{1 - 3 x^{4}}{{(1 + x^{4})}^{2}}$

Step 6

Combine $2$ and $\frac{1 - 3 x^{4}}{{(1 + x^{4})}^{2}}$ .

$\frac{2 (1 - 3 x^{4})}{{(1 + x^{4})}^{2}}$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$\frac{2 \cdot 1 + 2 (- 3 x^{4})}{{(1 + x^{4})}^{2}}$

Step 7.2

Simplify each term.

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Step 7.2.1

Multiply $2$ by $1$ .

$\frac{2 + 2 (- 3 x^{4})}{{(1 + x^{4})}^{2}}$

Step 7.2.2

Multiply $- 3$ by $2$ .

$\frac{2 - 6 x^{4}}{{(1 + x^{4})}^{2}}$

Step 7.3

Reorder terms.

$\frac{- 6 x^{4} + 2}{{(x^{4} + 1)}^{2}}$