Calculus Examples

$\log_{a} (x)$

Step 1

Rewrite $\log_{a} (x)$ using the change of base formula.

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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.

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

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

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

Step 2

Differentiate using the Constant Multiple Rule.

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

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

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

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

Step 3

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

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

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

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

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

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

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

Step 4

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

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

Step 5

Combine fractions.

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

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

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

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

Combine $\ln (x)$ and $\frac{1}{a \ln^{2} (a)}$ .

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