Calculus Examples

$y = 7 \ln (6 x)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (7 \ln (6 x))$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $7$ is constant with respect to $x$ , the derivative of $7 \ln (6 x)$ with respect to $x$ is $7 \frac{d}{d x} [\ln (6 x)]$ .

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

Step 3.2

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

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Step 3.2.1

To apply the Chain Rule, set $u$ as $6 x$ .

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

Step 3.2.2

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

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

Step 3.2.3

Replace all occurrences of $u$ with $6 x$ .

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

Step 3.3

Differentiate.

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Step 3.3.1

Combine $\frac{1}{6 x}$ and $7$ .

$\frac{7}{6 x} \frac{d}{d x} [6 x]$

Step 3.3.2

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

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

Step 3.3.3

Simplify terms.

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Step 3.3.3.1

Combine $6$ and $\frac{7}{6 x}$ .

$\frac{6 \cdot 7}{6 x} \frac{d}{d x} [x]$

Step 3.3.3.2

Multiply $6$ by $7$ .

$\frac{42}{6 x} \frac{d}{d x} [x]$

Step 3.3.3.3

Cancel the common factor of $42$ and $6$ .

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Step 3.3.3.3.1

Factor $6$ out of $42$ .

$\frac{6 \cdot 7}{6 x} \frac{d}{d x} [x]$

Step 3.3.3.3.2

Cancel the common factors.

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Step 3.3.3.3.2.1

Factor $6$ out of $6 x$ .

$\frac{6 \cdot 7}{6 (x)} \frac{d}{d x} [x]$

Step 3.3.3.3.2.2

Cancel the common factor.

$\frac{6 \cdot 7}{6 x} \frac{d}{d x} [x]$

Step 3.3.3.3.2.3

Rewrite the expression.

$\frac{7}{x} \frac{d}{d x} [x]$

Step 3.3.4

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

$\frac{7}{x} \cdot 1$

Step 3.3.5

Multiply $\frac{7}{x}$ by $1$ .

$\frac{7}{x}$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = \frac{7}{x}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{7}{x}$