Calculus Examples

$f (x) = - 4 \log_{2} (- 3 x)$

Step 1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 \log_{2} (- 3 x)$ with respect to $x$ is $- 4 \frac{d}{d x} [\log_{2} (- 3 x)]$ .

$- 4 \frac{d}{d x} [\log_{2} (- 3 x)]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \log_{2} (x)$ and $g (x) = - 3 x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $- 3 x$ .

$- 4 (\frac{d}{d u} [\log_{2} (u)] \frac{d}{d x} [- 3 x])$

Step 2.2

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

$- 4 (\frac{1}{u \ln (2)} \frac{d}{d x} [- 3 x])$

Step 2.3

Replace all occurrences of $u$ with $- 3 x$ .

$- 4 (\frac{1}{- 3 x \ln (2)} \frac{d}{d x} [- 3 x])$

Step 3

Differentiate.

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Step 3.1

Move the negative in front of the fraction.

$- 4 (- \frac{1}{3 x \ln (2)} \frac{d}{d x} [- 3 x])$

Step 3.2

Combine fractions.

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Step 3.2.1

Multiply $- 1$ by $- 4$ .

$4 (\frac{1}{3 x \ln (2)} \frac{d}{d x} [- 3 x])$

Step 3.2.2

Combine $\frac{1}{3 x \ln (2)}$ and $4$ .

$\frac{4}{3 x \ln (2)} \frac{d}{d x} [- 3 x]$

Step 3.3

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

$\frac{4}{3 x \ln (2)} (- 3 \frac{d}{d x} [x])$

Step 3.4

Simplify terms.

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Step 3.4.1

Combine $- 3$ and $\frac{4}{3 x \ln (2)}$ .

$\frac{- 3 \cdot 4}{3 x \ln (2)} \frac{d}{d x} [x]$

Step 3.4.2

Multiply $- 3$ by $4$ .

$\frac{- 12}{3 x \ln (2)} \frac{d}{d x} [x]$

Step 3.4.3

Cancel the common factor of $- 12$ and $3$ .

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Step 3.4.3.1

Factor $3$ out of $- 12$ .

$\frac{3 \cdot - 4}{3 x \ln (2)} \frac{d}{d x} [x]$

Step 3.4.3.2

Cancel the common factors.

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Step 3.4.3.2.1

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

$\frac{3 \cdot - 4}{3 (x \ln (2))} \frac{d}{d x} [x]$

Step 3.4.3.2.2

Cancel the common factor.

$\frac{3 \cdot - 4}{3 (x \ln (2))} \frac{d}{d x} [x]$

Step 3.4.3.2.3

Rewrite the expression.

$\frac{- 4}{x \ln (2)} \frac{d}{d x} [x]$

Step 3.4.4

Move the negative in front of the fraction.

$- \frac{4}{x \ln (2)} \frac{d}{d x} [x]$

Step 3.5

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

$- \frac{4}{x \ln (2)} \cdot 1$

Step 3.6

Multiply $- 1$ by $1$ .

$- \frac{4}{x \ln (2)}$