Calculus Examples

$y = \frac{1}{\ln (3 x)}$

Step 1

Rewrite $\frac{1}{\ln (3 x)}$ as $\ln^{- 1} (3 x)$ .

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

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 $g (x) = \ln (3 x)$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u_{1}$ with $\ln (3 x)$ .

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

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 $g (x) = 3 x$ .

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Step 3.1

To apply the Chain Rule, set $u_{2}$ as $3 x$ .

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

Step 3.2

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

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

Step 3.3

Replace all occurrences of $u_{2}$ with $3 x$ .

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

Step 4

Differentiate.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Simplify terms.

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Step 4.4.1

Multiply $3$ by $- 1$ .

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

Step 4.4.2

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

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

Step 4.4.3

Cancel the common factor of $- 3$ and $3$ .

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Step 4.4.3.1

Factor $3$ out of $- 3$ .

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

Step 4.4.3.2

Cancel the common factors.

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Step 4.4.3.2.1

Factor $3$ out of $3 x \ln^{2} (3 x)$ .

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

Step 4.4.3.2.2

Cancel the common factor.

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

Step 4.4.3.2.3

Rewrite the expression.

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

Step 4.4.4

Move the negative in front of the fraction.

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

Step 4.5

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

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

Step 4.6

Multiply $- 1$ by $1$ .

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