Calculus Examples

$\log_{b} (18)$

Step 1

Rewrite $\log_{b} (18)$ using the change of base formula.

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Step 1.1

The change of base rule can be used if $a$ and $b$ are greater than 0 and not equal to 1, and $x$ is greater than 0.

$\frac{d}{d b} [\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}]$

Step 1.2

Substitute in values for the variables in the change of base formula, using $b = e$ .

$\frac{d}{d b} [\frac{\ln (18)}{\ln (b)}]$

Step 2

Differentiate using the Constant Multiple Rule.

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Step 2.1

Since $\ln (18)$ is constant with respect to $b$ , the derivative of $\frac{\ln (18)}{\ln (b)}$ with respect to $b$ is $\ln (18) \frac{d}{d b} [\frac{1}{\ln (b)}]$ .

$\ln (18) \frac{d}{d b} [\frac{1}{\ln (b)}]$

Step 2.2

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

$\ln (18) \frac{d}{d b} [\ln^{- 1} (b)]$

Step 3

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

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Step 3.1

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

$\ln (18) (\frac{d}{d u} [u^{- 1}] \frac{d}{d b} [\ln (b)])$

Step 3.2

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

$\ln (18) (- u^{- 2} \frac{d}{d b} [\ln (b)])$

Step 3.3

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

$\ln (18) (- \ln^{- 2} (b) \frac{d}{d b} [\ln (b)])$

Step 4

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

$\ln (18) (- \ln^{- 2} (b) \frac{1}{b})$

Step 5

Combine fractions.

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Step 5.1

Combine $\frac{1}{b}$ and $\ln^{- 2} (b)$ .

$\ln (18) (- \frac{\ln^{- 2} (b)}{b})$

Step 5.2

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

$\ln (18) (- \frac{1}{b \ln^{2} (b)})$

Step 5.3

Combine $\ln (18)$ and $\frac{1}{b \ln^{2} (b)}$ .

$- \frac{\ln (18)}{b \ln^{2} (b)}$