Calculus Examples

$\log_{6} (7 w)$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d w} [f (g (w))]$ is $f' (g (w)) g' (w)$ where $f (w) = \log_{6} (w)$ and $g (w) = 7 w$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $7 w$ .

$\frac{d}{d u} [\log_{6} (u)] \frac{d}{d w} [7 w]$

Step 1.2

The derivative of $\log_{6} (u)$ with respect to $u$ is $\frac{1}{u \ln (6)}$ .

$\frac{1}{u \ln (6)} \frac{d}{d w} [7 w]$

Step 1.3

Replace all occurrences of $u$ with $7 w$ .

$\frac{1}{7 w \ln (6)} \frac{d}{d w} [7 w]$

Step 2

Differentiate.

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Step 2.1

Since $7$ is constant with respect to $w$ , the derivative of $7 w$ with respect to $w$ is $7 \frac{d}{d w} [w]$ .

$\frac{1}{7 w \ln (6)} (7 \frac{d}{d w} [w])$

Step 2.2

Simplify terms.

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Step 2.2.1

Combine $7$ and $\frac{1}{7 w \ln (6)}$ .

$\frac{7}{7 w \ln (6)} \frac{d}{d w} [w]$

Step 2.2.2

Cancel the common factor of $7$ .

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Step 2.2.2.1

Cancel the common factor.

$\frac{7}{7 w \ln (6)} \frac{d}{d w} [w]$

Step 2.2.2.2

Rewrite the expression.

$\frac{1}{w \ln (6)} \frac{d}{d w} [w]$

Step 2.3

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

$\frac{1}{w \ln (6)} \cdot 1$

Step 2.4

Multiply $\frac{1}{w \ln (6)}$ by $1$ .

$\frac{1}{w \ln (6)}$