Calculus Examples

$\frac{1}{7 x^{2} + 7}$

Step 1

Rewrite $\frac{1}{7 x^{2} + 7}$ as ${(7 x^{2} + 7)}^{- 1}$ .

$\frac{d}{d x} [{(7 x^{2} + 7)}^{- 1}]$

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 $g (x) = 7 x^{2} + 7$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $7 x^{2} + 7$ .

$\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [7 x^{2} + 7]$

Step 2.2

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

$- u^{- 2} \frac{d}{d x} [7 x^{2} + 7]$

Step 2.3

Replace all occurrences of $u$ with $7 x^{2} + 7$ .

$- {(7 x^{2} + 7)}^{- 2} \frac{d}{d x} [7 x^{2} + 7]$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $7 x^{2} + 7$ with respect to $x$ is $\frac{d}{d x} [7 x^{2}] + \frac{d}{d x} [7]$ .

$- {(7 x^{2} + 7)}^{- 2} (\frac{d}{d x} [7 x^{2}] + \frac{d}{d x} [7])$

Step 3.2

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{2}$ with respect to $x$ is $7 \frac{d}{d x} [x^{2}]$ .

$- {(7 x^{2} + 7)}^{- 2} (7 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [7])$

Step 3.3

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

$- {(7 x^{2} + 7)}^{- 2} (7 (2 x) + \frac{d}{d x} [7])$

Step 3.4

Multiply $2$ by $7$ .

$- {(7 x^{2} + 7)}^{- 2} (14 x + \frac{d}{d x} [7])$

Step 3.5

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$- {(7 x^{2} + 7)}^{- 2} (14 x + 0)$

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $14 x$ and $0$ .

$- {(7 x^{2} + 7)}^{- 2} (14 x)$

Step 3.6.2

Multiply $14$ by $- 1$ .

$- 14 {(7 x^{2} + 7)}^{- 2} x$

Step 4

Simplify.

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Step 4.1

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

$- 14 \frac{1}{{(7 x^{2} + 7)}^{2}} x$

Step 4.2

Combine terms.

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Step 4.2.1

Combine $- 14$ and $\frac{1}{{(7 x^{2} + 7)}^{2}}$ .

$\frac{- 14}{{(7 x^{2} + 7)}^{2}} x$

Step 4.2.2

Move the negative in front of the fraction.

$- \frac{14}{{(7 x^{2} + 7)}^{2}} x$

Step 4.2.3

Combine $x$ and $\frac{14}{{(7 x^{2} + 7)}^{2}}$ .

$- \frac{x \cdot 14}{{(7 x^{2} + 7)}^{2}}$

Step 4.2.4

Move $14$ to the left of $x$ .

$- \frac{14 x}{{(7 x^{2} + 7)}^{2}}$

Step 4.3

Simplify the denominator.

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Step 4.3.1

Factor $7$ out of $7 x^{2} + 7$ .

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Step 4.3.1.1

Factor $7$ out of $7 x^{2}$ .

$- \frac{14 x}{{(7 (x^{2}) + 7)}^{2}}$

Step 4.3.1.2

Factor $7$ out of $7$ .

$- \frac{14 x}{{(7 (x^{2}) + 7 (1))}^{2}}$

Step 4.3.1.3

Factor $7$ out of $7 (x^{2}) + 7 (1)$ .

$- \frac{14 x}{{(7 (x^{2} + 1))}^{2}}$

Step 4.3.2

Apply the product rule to $7 (x^{2} + 1)$ .

$- \frac{14 x}{7^{2} {(x^{2} + 1)}^{2}}$

Step 4.3.3

Raise $7$ to the power of $2$ .

$- \frac{14 x}{49 {(x^{2} + 1)}^{2}}$

Step 4.4

Cancel the common factor of $14$ and $49$ .

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Step 4.4.1

Factor $7$ out of $14 x$ .

$- \frac{7 (2 x)}{49 {(x^{2} + 1)}^{2}}$

Step 4.4.2

Cancel the common factors.

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Step 4.4.2.1

Factor $7$ out of $49 {(x^{2} + 1)}^{2}$ .

$- \frac{7 (2 x)}{7 (7 {(x^{2} + 1)}^{2})}$

Step 4.4.2.2

Cancel the common factor.

$- \frac{7 (2 x)}{7 (7 {(x^{2} + 1)}^{2})}$

Step 4.4.2.3

Rewrite the expression.

$- \frac{2 x}{7 {(x^{2} + 1)}^{2}}$