Calculus Examples

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

Step 1

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

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

Step 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 $g (x) = 4 x^{2} + 4$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $4 x^{2} + 4$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $4 x^{2} + 4$ .

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

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $4 x^{2} + 4$ with respect to $x$ is $\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [4]$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Multiply $2$ by $4$ .

$- {(4 x^{2} + 4)}^{- 2} (8 x + \frac{d}{d x} [4])$

Step 3.5

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$- {(4 x^{2} + 4)}^{- 2} (8 x + 0)$

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $8 x$ and $0$ .

$- {(4 x^{2} + 4)}^{- 2} (8 x)$

Step 3.6.2

Multiply $8$ by $- 1$ .

$- 8 {(4 x^{2} + 4)}^{- 2} x$

Step 4

Simplify.

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Step 4.1

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

$- 8 \frac{1}{{(4 x^{2} + 4)}^{2}} x$

Step 4.2

Combine terms.

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Step 4.2.1

Combine $- 8$ and $\frac{1}{{(4 x^{2} + 4)}^{2}}$ .

$\frac{- 8}{{(4 x^{2} + 4)}^{2}} x$

Step 4.2.2

Move the negative in front of the fraction.

$- \frac{8}{{(4 x^{2} + 4)}^{2}} x$

Step 4.2.3

Combine $x$ and $\frac{8}{{(4 x^{2} + 4)}^{2}}$ .

$- \frac{x \cdot 8}{{(4 x^{2} + 4)}^{2}}$

Step 4.2.4

Move $8$ to the left of $x$ .

$- \frac{8 x}{{(4 x^{2} + 4)}^{2}}$

Step 4.3

Simplify the denominator.

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Step 4.3.1

Factor $4$ out of $4 x^{2} + 4$ .

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Step 4.3.1.1

Factor $4$ out of $4 x^{2}$ .

$- \frac{8 x}{{(4 (x^{2}) + 4)}^{2}}$

Step 4.3.1.2

Factor $4$ out of $4$ .

$- \frac{8 x}{{(4 (x^{2}) + 4 (1))}^{2}}$

Step 4.3.1.3

Factor $4$ out of $4 (x^{2}) + 4 (1)$ .

$- \frac{8 x}{{(4 (x^{2} + 1))}^{2}}$

Step 4.3.2

Apply the product rule to $4 (x^{2} + 1)$ .

$- \frac{8 x}{4^{2} {(x^{2} + 1)}^{2}}$

Step 4.3.3

Raise $4$ to the power of $2$ .

$- \frac{8 x}{16 {(x^{2} + 1)}^{2}}$

Step 4.4

Cancel the common factor of $8$ and $16$ .

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Step 4.4.1

Factor $8$ out of $8 x$ .

$- \frac{8 (x)}{16 {(x^{2} + 1)}^{2}}$

Step 4.4.2

Cancel the common factors.

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Step 4.4.2.1

Factor $8$ out of $16 {(x^{2} + 1)}^{2}$ .

$- \frac{8 (x)}{8 (2 {(x^{2} + 1)}^{2})}$

Step 4.4.2.2

Cancel the common factor.

$- \frac{8 x}{8 (2 {(x^{2} + 1)}^{2})}$

Step 4.4.2.3

Rewrite the expression.

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