Calculus Examples

$y = \frac{\ln (x)}{x^{4}}$

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 $g (x) = x^{4}$ .

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

Step 2

Multiply the exponents in ${(x^{4})}^{2}$ .

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Step 2.1

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

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

Step 2.2

Multiply $4$ by $2$ .

$\frac{x^{4} \frac{d}{d x} [\ln (x)] - \ln (x) \frac{d}{d x} [x^{4}]}{x^{8}}$

Step 3

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

$\frac{x^{4} \frac{1}{x} - \ln (x) \frac{d}{d x} [x^{4}]}{x^{8}}$

Step 4

Differentiate using the Power Rule.

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Step 4.1

Combine $x^{4}$ and $\frac{1}{x}$ .

$\frac{\frac{x^{4}}{x} - \ln (x) \frac{d}{d x} [x^{4}]}{x^{8}}$

Step 4.2

Cancel the common factor of $x^{4}$ and $x$ .

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Step 4.2.1

Factor $x$ out of $x^{4}$ .

$\frac{\frac{x \cdot x^{3}}{x} - \ln (x) \frac{d}{d x} [x^{4}]}{x^{8}}$

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

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

$\frac{\frac{x \cdot x^{3}}{x^{1}} - \ln (x) \frac{d}{d x} [x^{4}]}{x^{8}}$

Step 4.2.2.2

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

$\frac{\frac{x \cdot x^{3}}{x \cdot 1} - \ln (x) \frac{d}{d x} [x^{4}]}{x^{8}}$

Step 4.2.2.3

Cancel the common factor.

$\frac{\frac{x \cdot x^{3}}{x \cdot 1} - \ln (x) \frac{d}{d x} [x^{4}]}{x^{8}}$

Step 4.2.2.4

Rewrite the expression.

$\frac{\frac{x^{3}}{1} - \ln (x) \frac{d}{d x} [x^{4}]}{x^{8}}$

Step 4.2.2.5

Divide $x^{3}$ by $1$ .

$\frac{x^{3} - \ln (x) \frac{d}{d x} [x^{4}]}{x^{8}}$

Step 4.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{x^{3} - \ln (x) (4 x^{3})}{x^{8}}$

Step 4.4

Simplify with factoring out.

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Step 4.4.1

Multiply $4$ by $- 1$ .

$\frac{x^{3} - 4 \ln (x) x^{3}}{x^{8}}$

Step 4.4.2

Factor $x^{3}$ out of $x^{3} - 4 \ln (x) x^{3}$ .

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Step 4.4.2.1

Multiply by $1$ .

$\frac{x^{3} \cdot 1 - 4 \ln (x) x^{3}}{x^{8}}$

Step 4.4.2.2

Factor $x^{3}$ out of $- 4 \ln (x) x^{3}$ .

$\frac{x^{3} \cdot 1 + x^{3} (- 4 \ln (x))}{x^{8}}$

Step 4.4.2.3

Factor $x^{3}$ out of $x^{3} \cdot 1 + x^{3} (- 4 \ln (x))$ .

$\frac{x^{3} (1 - 4 \ln (x))}{x^{8}}$

Step 5

Cancel the common factors.

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Step 5.1

Factor $x^{3}$ out of $x^{8}$ .

$\frac{x^{3} (1 - 4 \ln (x))}{x^{3} x^{5}}$

Step 5.2

Cancel the common factor.

$\frac{x^{3} (1 - 4 \ln (x))}{x^{3} x^{5}}$

Step 5.3

Rewrite the expression.

$\frac{1 - 4 \ln (x)}{x^{5}}$

Step 6

Simplify $- 4 \ln (x)$ by moving $4$ inside the logarithm.

$\frac{1 - \ln (x^{4})}{x^{5}}$