Calculus Examples

$f (x) = 4 \log_{7} (\sqrt{x} - 2)$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\frac{d}{d x} [4 \log_{7} (x^{\frac{1}{2}} - 2)]$

Step 1.2

Since $4$ is constant with respect to $x$ , the derivative of $4 \log_{7} (x^{\frac{1}{2}} - 2)$ with respect to $x$ is $4 \frac{d}{d x} [\log_{7} (x^{\frac{1}{2}} - 2)]$ .

$4 \frac{d}{d x} [\log_{7} (x^{\frac{1}{2}} - 2)]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \log_{7} (x)$ and $g (x) = x^{\frac{1}{2}} - 2$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $x^{\frac{1}{2}} - 2$ .

$4 (\frac{d}{d u} [\log_{7} (u)] \frac{d}{d x} [x^{\frac{1}{2}} - 2])$

Step 2.2

The derivative of $\log_{7} (u)$ with respect to $u$ is $\frac{1}{u \ln (7)}$ .

$4 (\frac{1}{u \ln (7)} \frac{d}{d x} [x^{\frac{1}{2}} - 2])$

Step 2.3

Replace all occurrences of $u$ with $x^{\frac{1}{2}} - 2$ .

$4 (\frac{1}{(x^{\frac{1}{2}} - 2) \ln (7)} \frac{d}{d x} [x^{\frac{1}{2}} - 2])$

Step 3

Differentiate.

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Step 3.1

Combine $\frac{1}{(x^{\frac{1}{2}} - 2) \ln (7)}$ and $4$ .

$\frac{4}{(x^{\frac{1}{2}} - 2) \ln (7)} \frac{d}{d x} [x^{\frac{1}{2}} - 2]$

Step 3.2

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

$\frac{4}{(x^{\frac{1}{2}} - 2) \ln (7)} (\frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [- 2])$

Step 3.3

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

$\frac{4}{(x^{\frac{1}{2}} - 2) \ln (7)} (\frac{1}{2} x^{\frac{1}{2} - 1} + \frac{d}{d x} [- 2])$

Step 4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{4}{(x^{\frac{1}{2}} - 2) \ln (7)} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}} + \frac{d}{d x} [- 2])$

Step 5

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{4}{(x^{\frac{1}{2}} - 2) \ln (7)} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} + \frac{d}{d x} [- 2])$

Step 6

Combine the numerators over the common denominator.

$\frac{4}{(x^{\frac{1}{2}} - 2) \ln (7)} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}} + \frac{d}{d x} [- 2])$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

$\frac{4}{(x^{\frac{1}{2}} - 2) \ln (7)} (\frac{1}{2} x^{\frac{1 - 2}{2}} + \frac{d}{d x} [- 2])$

Step 7.2

Subtract $2$ from $1$ .

$\frac{4}{(x^{\frac{1}{2}} - 2) \ln (7)} (\frac{1}{2} x^{\frac{- 1}{2}} + \frac{d}{d x} [- 2])$

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

$\frac{4}{(x^{\frac{1}{2}} - 2) \ln (7)} (\frac{1}{2} x^{- \frac{1}{2}} + \frac{d}{d x} [- 2])$

Step 8.2

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

$\frac{4}{(x^{\frac{1}{2}} - 2) \ln (7)} (\frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- 2])$

Step 8.3

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{4}{(x^{\frac{1}{2}} - 2) \ln (7)} (\frac{1}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- 2])$

Step 9

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$\frac{4}{(x^{\frac{1}{2}} - 2) \ln (7)} (\frac{1}{2 x^{\frac{1}{2}}} + 0)$

Step 10

Simplify terms.

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Step 10.1

Add $\frac{1}{2 x^{\frac{1}{2}}}$ and $0$ .

$\frac{4}{(x^{\frac{1}{2}} - 2) \ln (7)} \cdot \frac{1}{2 x^{\frac{1}{2}}}$

Step 10.2

Multiply $\frac{4}{(x^{\frac{1}{2}} - 2) \ln (7)}$ by $\frac{1}{2 x^{\frac{1}{2}}}$ .

$\frac{4}{(x^{\frac{1}{2}} - 2) \ln (7) (2 x^{\frac{1}{2}})}$

Step 10.3

Factor $2$ out of $4$ .

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

Step 11

Cancel the common factors.

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Step 11.1

Factor $2$ out of $(x^{\frac{1}{2}} - 2) \ln (7) (2 x^{\frac{1}{2}})$ .

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

Step 11.2

Cancel the common factor.

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

Step 11.3

Rewrite the expression.

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

Step 12

Simplify.

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Step 12.1

Apply the distributive property.

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

Step 12.2

Apply the distributive property.

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

Step 12.3

Combine terms.

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Step 12.3.1

Multiply $x^{\frac{1}{2}}$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 12.3.1.1

Move $x^{\frac{1}{2}}$ .

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

Step 12.3.1.2

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

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

Step 12.3.1.3

Combine the numerators over the common denominator.

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

Step 12.3.1.4

Add $1$ and $1$ .

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

Step 12.3.1.5

Divide $2$ by $2$ .

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

Step 12.3.2

Simplify $x^{1} \ln (7)$ .

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

Step 12.4

Reorder terms.

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

Step 12.5

Factor $x^{\frac{1}{2}} \ln (7)$ out of $x \ln (7) - 2 x^{\frac{1}{2}} \ln (7)$ .

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Step 12.5.1

Factor $x^{\frac{1}{2}} \ln (7)$ out of $x \ln (7)$ .

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

Step 12.5.2

Factor $x^{\frac{1}{2}} \ln (7)$ out of $- 2 x^{\frac{1}{2}} \ln (7)$ .

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

Step 12.5.3

Factor $x^{\frac{1}{2}} \ln (7)$ out of $x^{\frac{1}{2}} \ln (7) (x^{\frac{1}{2}}) + x^{\frac{1}{2}} \ln (7) (- 2)$ .

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