Calculus Examples

$f (x) = \ln (\ln ({(5 x^{- 3})}^{\frac{1}{2}}))$

Step 1

Simplify with factoring out.

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Step 1.1

Factor $5$ out of $5 x^{- 3}$ .

$\frac{d}{d x} [\ln (\ln ({(5 (x^{- 3}))}^{\frac{1}{2}}))]$

Step 1.2

Apply basic rules of exponents.

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Step 1.2.1

Apply the product rule to $5 (x^{- 3})$ .

$\frac{d}{d x} [\ln (\ln (5^{\frac{1}{2}} {(x^{- 3})}^{\frac{1}{2}}))]$

Step 1.2.2

Multiply the exponents in ${(x^{- 3})}^{\frac{1}{2}}$ .

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Step 1.2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{d}{d x} [\ln (\ln (5^{\frac{1}{2}} x^{- 3 (\frac{1}{2})}))]$

Step 1.2.2.2

Combine $- 3$ and $\frac{1}{2}$ .

$\frac{d}{d x} [\ln (\ln (5^{\frac{1}{2}} x^{\frac{- 3}{2}}))]$

Step 1.2.2.3

Move the negative in front of the fraction.

$\frac{d}{d x} [\ln (\ln (5^{\frac{1}{2}} x^{- \frac{3}{2}}))]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = \ln (5^{\frac{1}{2}} x^{- \frac{3}{2}})$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $\ln (5^{\frac{1}{2}} x^{- \frac{3}{2}})$ .

$\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [\ln (5^{\frac{1}{2}} x^{- \frac{3}{2}})]$

Step 2.2

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

$\frac{1}{u_{1}} \frac{d}{d x} [\ln (5^{\frac{1}{2}} x^{- \frac{3}{2}})]$

Step 2.3

Replace all occurrences of $u_{1}$ with $\ln (5^{\frac{1}{2}} x^{- \frac{3}{2}})$ .

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Step 2.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{\ln (5^{\frac{1}{2}} x^{- 3 (\frac{1}{2})})} \frac{d}{d x} [\ln (5^{\frac{1}{2}} x^{- \frac{3}{2}})]$

Step 2.3.2

Combine $- 3$ and $\frac{1}{2}$ .

$\frac{1}{\ln (5^{\frac{1}{2}} x^{\frac{- 3}{2}})} \frac{d}{d x} [\ln (5^{\frac{1}{2}} x^{- \frac{3}{2}})]$

Step 2.3.3

Move the negative in front of the fraction.

$\frac{1}{\ln (5^{\frac{1}{2}} x^{- \frac{3}{2}})} \frac{d}{d x} [\ln (5^{\frac{1}{2}} x^{- \frac{3}{2}})]$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = 5^{\frac{1}{2}} x^{- \frac{3}{2}}$ .

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Step 3.1

To apply the Chain Rule, set $u_{2}$ as $5^{\frac{1}{2}} x^{- \frac{3}{2}}$ .

$\frac{1}{\ln (5^{\frac{1}{2}} x^{- \frac{3}{2}})} (\frac{d}{d u_{2}} [\ln (u_{2})] \frac{d}{d x} [5^{\frac{1}{2}} x^{- \frac{3}{2}}])$

Step 3.2

The derivative of $\ln (u_{2})$ with respect to $u_{2}$ is $\frac{1}{u_{2}}$ .

$\frac{1}{\ln (5^{\frac{1}{2}} x^{- \frac{3}{2}})} (\frac{1}{u_{2}} \frac{d}{d x} [5^{\frac{1}{2}} x^{- \frac{3}{2}}])$

Step 3.3

Replace all occurrences of $u_{2}$ with $5^{\frac{1}{2}} x^{- \frac{3}{2}}$ .

$\frac{1}{\ln (5^{\frac{1}{2}} x^{- \frac{3}{2}})} (\frac{1}{5^{\frac{1}{2}} x^{- \frac{3}{2}}} \frac{d}{d x} [5^{\frac{1}{2}} x^{- \frac{3}{2}}])$

Step 4

Differentiate.

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Step 4.1

Move $x^{- \frac{3}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{1}{\ln (5^{\frac{1}{2}} x^{- \frac{3}{2}})} (\frac{x^{\frac{3}{2}}}{5^{\frac{1}{2}}} \frac{d}{d x} [5^{\frac{1}{2}} x^{- \frac{3}{2}}])$

Step 4.2

Multiply $\frac{x^{\frac{3}{2}}}{5^{\frac{1}{2}}}$ by $\frac{1}{\ln (5^{\frac{1}{2}} x^{- \frac{3}{2}})}$ .

$\frac{x^{\frac{3}{2}}}{5^{\frac{1}{2}} \ln (5^{\frac{1}{2}} x^{- \frac{3}{2}})} \frac{d}{d x} [5^{\frac{1}{2}} x^{- \frac{3}{2}}]$

Step 4.3

Since $5^{\frac{1}{2}}$ is constant with respect to $x$ , the derivative of $5^{\frac{1}{2}} x^{- \frac{3}{2}}$ with respect to $x$ is $5^{\frac{1}{2}} \frac{d}{d x} [x^{- \frac{3}{2}}]$ .

$\frac{x^{\frac{3}{2}}}{5^{\frac{1}{2}} \ln (5^{\frac{1}{2}} x^{- \frac{3}{2}})} (5^{\frac{1}{2}} \frac{d}{d x} [x^{- \frac{3}{2}}])$

Step 4.4

Simplify terms.

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Step 4.4.1

Combine $5^{\frac{1}{2}}$ and $\frac{x^{\frac{3}{2}}}{5^{\frac{1}{2}} \ln (5^{\frac{1}{2}} x^{- \frac{3}{2}})}$ .

$\frac{5^{\frac{1}{2}} x^{\frac{3}{2}}}{5^{\frac{1}{2}} \ln (5^{\frac{1}{2}} x^{- \frac{3}{2}})} \frac{d}{d x} [x^{- \frac{3}{2}}]$

Step 4.4.2

Cancel the common factor.

$\frac{5^{\frac{1}{2}} x^{\frac{3}{2}}}{5^{\frac{1}{2}} \ln (5^{\frac{1}{2}} x^{- \frac{3}{2}})} \frac{d}{d x} [x^{- \frac{3}{2}}]$

Step 4.4.3

Rewrite the expression.

$\frac{x^{\frac{3}{2}}}{\ln (5^{\frac{1}{2}} x^{- \frac{3}{2}})} \frac{d}{d x} [x^{- \frac{3}{2}}]$

Step 4.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{3}{2}$ .

$\frac{x^{\frac{3}{2}}}{\ln (5^{\frac{1}{2}} x^{- \frac{3}{2}})} (- \frac{3}{2} x^{- \frac{3}{2} - 1})$

Step 5

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

$\frac{x^{\frac{3}{2}}}{\ln (5^{\frac{1}{2}} x^{- \frac{3}{2}})} (- \frac{3}{2} x^{- \frac{3}{2} - 1 \cdot \frac{2}{2}})$

Step 6

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

$\frac{x^{\frac{3}{2}}}{\ln (5^{\frac{1}{2}} x^{- \frac{3}{2}})} (- \frac{3}{2} x^{- \frac{3}{2} + \frac{- 1 \cdot 2}{2}})$

Step 7

Combine the numerators over the common denominator.

$\frac{x^{\frac{3}{2}}}{\ln (5^{\frac{1}{2}} x^{- \frac{3}{2}})} (- \frac{3}{2} x^{\frac{- 3 - 1 \cdot 2}{2}})$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $2$ .

$\frac{x^{\frac{3}{2}}}{\ln (5^{\frac{1}{2}} x^{- \frac{3}{2}})} (- \frac{3}{2} x^{\frac{- 3 - 2}{2}})$

Step 8.2

Subtract $2$ from $- 3$ .

$\frac{x^{\frac{3}{2}}}{\ln (5^{\frac{1}{2}} x^{- \frac{3}{2}})} (- \frac{3}{2} x^{\frac{- 5}{2}})$

Step 9

Move the negative in front of the fraction.

$\frac{x^{\frac{3}{2}}}{\ln (5^{\frac{1}{2}} x^{- \frac{3}{2}})} (- \frac{3}{2} x^{- \frac{5}{2}})$

Step 10

Combine $x^{- \frac{5}{2}}$ and $\frac{3}{2}$ .

$\frac{x^{\frac{3}{2}}}{\ln (5^{\frac{1}{2}} x^{- \frac{3}{2}})} (- \frac{x^{- \frac{5}{2}} \cdot 3}{2})$

Step 11

Multiply $\frac{x^{\frac{3}{2}}}{\ln (5^{\frac{1}{2}} x^{- \frac{3}{2}})}$ by $\frac{x^{- \frac{5}{2}} \cdot 3}{2}$ .

$- \frac{x^{\frac{3}{2}} (x^{- \frac{5}{2}} \cdot 3)}{\ln (5^{\frac{1}{2}} x^{- \frac{3}{2}}) \cdot 2}$

Step 12

Multiply $x^{\frac{3}{2}}$ by $x^{- \frac{5}{2}}$ by adding the exponents.

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Step 12.1

Move $x^{- \frac{5}{2}}$ .

$- \frac{x^{- \frac{5}{2}} x^{\frac{3}{2}} \cdot 3}{\ln (5^{\frac{1}{2}} x^{- \frac{3}{2}}) \cdot 2}$

Step 12.2

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

$- \frac{x^{- \frac{5}{2} + \frac{3}{2}} \cdot 3}{\ln (5^{\frac{1}{2}} x^{- \frac{3}{2}}) \cdot 2}$

Step 12.3

Combine the numerators over the common denominator.

$- \frac{x^{\frac{- 5 + 3}{2}} \cdot 3}{\ln (5^{\frac{1}{2}} x^{- \frac{3}{2}}) \cdot 2}$

Step 12.4

Add $- 5$ and $3$ .

$- \frac{x^{\frac{- 2}{2}} \cdot 3}{\ln (5^{\frac{1}{2}} x^{- \frac{3}{2}}) \cdot 2}$

Step 12.5

Divide $- 2$ by $2$ .

$- \frac{x^{- 1} \cdot 3}{\ln (5^{\frac{1}{2}} x^{- \frac{3}{2}}) \cdot 2}$

Step 13

Move $3$ to the left of $x^{- 1}$ .

$- \frac{3 \cdot x^{- 1}}{\ln (5^{\frac{1}{2}} x^{- \frac{3}{2}}) \cdot 2}$

Step 14

Move $2$ to the left of $\ln (5^{\frac{1}{2}} x^{- \frac{3}{2}})$ .

$- \frac{3 x^{- 1}}{2 \cdot \ln (5^{\frac{1}{2}} x^{- \frac{3}{2}})}$

Step 15

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

$- \frac{3}{2 \ln (5^{\frac{1}{2}} x^{- \frac{3}{2}}) x}$

Step 16

Simplify $2 \ln (5^{\frac{1}{2}} x^{- \frac{3}{2}})$ by moving $2$ inside the logarithm.

$- \frac{3}{\ln ({(5^{\frac{1}{2}} x^{- \frac{3}{2}})}^{2}) x}$

Step 17

Simplify.

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Step 17.1

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

$- \frac{3}{\ln ({(5^{\frac{1}{2}} \frac{1}{x^{\frac{3}{2}}})}^{2}) x}$

Step 17.2

Apply the product rule to $5^{\frac{1}{2}} \frac{1}{x^{\frac{3}{2}}}$ .

$- \frac{3}{\ln ({(5^{\frac{1}{2}})}^{2} {(\frac{1}{x^{\frac{3}{2}}})}^{2}) x}$

Step 17.3

Apply the product rule to $\frac{1}{x^{\frac{3}{2}}}$ .

$- \frac{3}{\ln ({(5^{\frac{1}{2}})}^{2} \frac{1^{2}}{{(x^{\frac{3}{2}})}^{2}}) x}$

Step 17.4

Combine terms.

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Step 17.4.1

Multiply the exponents in ${(5^{\frac{1}{2}})}^{2}$ .

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Step 17.4.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{3}{\ln (5^{\frac{1}{2} \cdot 2} \frac{1^{2}}{{(x^{\frac{3}{2}})}^{2}}) x}$

Step 17.4.1.2

Cancel the common factor of $2$ .

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Step 17.4.1.2.1

Cancel the common factor.

$- \frac{3}{\ln (5^{\frac{1}{2} \cdot 2} \frac{1^{2}}{{(x^{\frac{3}{2}})}^{2}}) x}$

Step 17.4.1.2.2

Rewrite the expression.

$- \frac{3}{\ln (5^{1} \frac{1^{2}}{{(x^{\frac{3}{2}})}^{2}}) x}$

Step 17.4.2

Evaluate the exponent.

$- \frac{3}{\ln (5 \frac{1^{2}}{{(x^{\frac{3}{2}})}^{2}}) x}$

Step 17.4.3

One to any power is one.

$- \frac{3}{\ln (5 \frac{1}{{(x^{\frac{3}{2}})}^{2}}) x}$

Step 17.4.4

Multiply the exponents in ${(x^{\frac{3}{2}})}^{2}$ .

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Step 17.4.4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{3}{\ln (5 \frac{1}{x^{\frac{3}{2} \cdot 2}}) x}$

Step 17.4.4.2

Cancel the common factor of $2$ .

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Step 17.4.4.2.1

Cancel the common factor.

$- \frac{3}{\ln (5 \frac{1}{x^{\frac{3}{2} \cdot 2}}) x}$

Step 17.4.4.2.2

Rewrite the expression.

$- \frac{3}{\ln (5 \frac{1}{x^{3}}) x}$

Step 17.4.5

Combine $5$ and $\frac{1}{x^{3}}$ .

$- \frac{3}{\ln (\frac{5}{x^{3}}) x}$

Step 17.5

Reorder factors in $- \frac{3}{\ln (\frac{5}{x^{3}}) x}$ .

$- \frac{3}{x \ln (\frac{5}{x^{3}})}$