Calculus Examples

$y = \frac{1}{\log_{2} (x)}$

Step 1

Rewrite $\frac{1}{\log_{2} (x)}$ as ${\log_{2}}^{- 1} (x)$ .

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

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 $g (x) = \log_{2} (x)$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $\log_{2} (x)$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $\log_{2} (x)$ .

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

Step 3

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

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

Step 4

Combine fractions.

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Step 4.1

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

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

Step 4.2

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

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