Calculus Examples

$\frac{9}{\ln (x)}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Since $9$ is constant with respect to $x$ , the derivative of $\frac{9}{\ln (x)}$ with respect to $x$ is $9 \frac{d}{d x} [\frac{1}{\ln (x)}]$ .

$9 \frac{d}{d x} [\frac{1}{\ln (x)}]$

Step 1.2

Rewrite $\frac{1}{\ln (x)}$ as $\ln^{- 1} (x)$ .

$9 \frac{d}{d x} [\ln^{- 1} (x)]$

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 $g (x) = \ln (x)$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $\ln (x)$ .

$9 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [\ln (x)])$

Step 2.2

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

$9 (- u^{- 2} \frac{d}{d x} [\ln (x)])$

Step 2.3

Replace all occurrences of $u$ with $\ln (x)$ .

$9 (- \ln^{- 2} (x) \frac{d}{d x} [\ln (x)])$

Step 3

Multiply $- 1$ by $9$ .

$- 9 (\ln^{- 2} (x) \frac{d}{d x} [\ln (x)])$

Step 4

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

$- 9 \ln^{- 2} (x) \frac{1}{x}$

Step 5

Combine fractions.

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Step 5.1

Combine $\frac{1}{x}$ and $- 9$ .

$\frac{- 9}{x} \ln^{- 2} (x)$

Step 5.2

Combine $\frac{- 9}{x}$ and $\ln^{- 2} (x)$ .

$\frac{- 9 \ln^{- 2} (x)}{x}$

Step 5.3

Simplify the expression.

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Step 5.3.1

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

$\frac{- 9}{x \ln^{2} (x)}$

Step 5.3.2

Move the negative in front of the fraction.

$- \frac{9}{x \ln^{2} (x)}$