Calculus Examples

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

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = \ln (x)$ and $g (x) = x^{6}$ .

$\frac{x^{6} \frac{d}{d x} [\ln (x)] - \ln (x) \frac{d}{d x} [x^{6}]}{{(x^{6})}^{2}}$

Step 2

Multiply the exponents in ${(x^{6})}^{2}$ .

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Step 2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{x^{6} \frac{d}{d x} [\ln (x)] - \ln (x) \frac{d}{d x} [x^{6}]}{x^{6 \cdot 2}}$

Step 2.2

Multiply $6$ by $2$ .

$\frac{x^{6} \frac{d}{d x} [\ln (x)] - \ln (x) \frac{d}{d x} [x^{6}]}{x^{12}}$

Step 3

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

$\frac{x^{6} \frac{1}{x} - \ln (x) \frac{d}{d x} [x^{6}]}{x^{12}}$

Step 4

Differentiate using the Power Rule.

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Step 4.1

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

$\frac{\frac{x^{6}}{x} - \ln (x) \frac{d}{d x} [x^{6}]}{x^{12}}$

Step 4.2

Cancel the common factor of $x^{6}$ and $x$ .

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Step 4.2.1

Factor $x$ out of $x^{6}$ .

$\frac{\frac{x \cdot x^{5}}{x} - \ln (x) \frac{d}{d x} [x^{6}]}{x^{12}}$

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Raise $x$ to the power of $1$ .

$\frac{\frac{x \cdot x^{5}}{x^{1}} - \ln (x) \frac{d}{d x} [x^{6}]}{x^{12}}$

Step 4.2.2.2

Factor $x$ out of $x^{1}$ .

$\frac{\frac{x \cdot x^{5}}{x \cdot 1} - \ln (x) \frac{d}{d x} [x^{6}]}{x^{12}}$

Step 4.2.2.3

Cancel the common factor.

$\frac{\frac{x \cdot x^{5}}{x \cdot 1} - \ln (x) \frac{d}{d x} [x^{6}]}{x^{12}}$

Step 4.2.2.4

Rewrite the expression.

$\frac{\frac{x^{5}}{1} - \ln (x) \frac{d}{d x} [x^{6}]}{x^{12}}$

Step 4.2.2.5

Divide $x^{5}$ by $1$ .

$\frac{x^{5} - \ln (x) \frac{d}{d x} [x^{6}]}{x^{12}}$

Step 4.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$\frac{x^{5} - \ln (x) (6 x^{5})}{x^{12}}$

Step 4.4

Simplify with factoring out.

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Step 4.4.1

Multiply $6$ by $- 1$ .

$\frac{x^{5} - 6 \ln (x) x^{5}}{x^{12}}$

Step 4.4.2

Factor $x^{5}$ out of $x^{5} - 6 \ln (x) x^{5}$ .

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Step 4.4.2.1

Multiply by $1$ .

$\frac{x^{5} \cdot 1 - 6 \ln (x) x^{5}}{x^{12}}$

Step 4.4.2.2

Factor $x^{5}$ out of $- 6 \ln (x) x^{5}$ .

$\frac{x^{5} \cdot 1 + x^{5} (- 6 \ln (x))}{x^{12}}$

Step 4.4.2.3

Factor $x^{5}$ out of $x^{5} \cdot 1 + x^{5} (- 6 \ln (x))$ .

$\frac{x^{5} (1 - 6 \ln (x))}{x^{12}}$

Step 5

Cancel the common factors.

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Step 5.1

Factor $x^{5}$ out of $x^{12}$ .

$\frac{x^{5} (1 - 6 \ln (x))}{x^{5} x^{7}}$

Step 5.2

Cancel the common factor.

$\frac{x^{5} (1 - 6 \ln (x))}{x^{5} x^{7}}$

Step 5.3

Rewrite the expression.

$\frac{1 - 6 \ln (x)}{x^{7}}$

Step 6

Simplify $- 6 \ln (x)$ by moving $6$ inside the logarithm.

$\frac{1 - \ln (x^{6})}{x^{7}}$