Calculus Examples

$\frac{8 \ln (x)}{x^{8}}$

Step 1

Since $8$ is constant with respect to $x$ , the derivative of $\frac{8 \ln (x)}{x^{8}}$ with respect to $x$ is $8 \frac{d}{d x} [\frac{\ln (x)}{x^{8}}]$ .

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

Step 2

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 $g (x) = x^{8}$ .

$8 \frac{x^{8} \frac{d}{d x} [\ln (x)] - \ln (x) \frac{d}{d x} [x^{8}]}{{(x^{8})}^{2}}$

Step 3

Multiply the exponents in ${(x^{8})}^{2}$ .

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Step 3.1

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

$8 \frac{x^{8} \frac{d}{d x} [\ln (x)] - \ln (x) \frac{d}{d x} [x^{8}]}{x^{8 \cdot 2}}$

Step 3.2

Multiply $8$ by $2$ .

$8 \frac{x^{8} \frac{d}{d x} [\ln (x)] - \ln (x) \frac{d}{d x} [x^{8}]}{x^{16}}$

Step 4

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

$8 \frac{x^{8} \frac{1}{x} - \ln (x) \frac{d}{d x} [x^{8}]}{x^{16}}$

Step 5

Differentiate using the Power Rule.

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Step 5.1

Combine $x^{8}$ and $\frac{1}{x}$ .

$8 \frac{\frac{x^{8}}{x} - \ln (x) \frac{d}{d x} [x^{8}]}{x^{16}}$

Step 5.2

Cancel the common factor of $x^{8}$ and $x$ .

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Step 5.2.1

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

$8 \frac{\frac{x \cdot x^{7}}{x} - \ln (x) \frac{d}{d x} [x^{8}]}{x^{16}}$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

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

$8 \frac{\frac{x \cdot x^{7}}{x^{1}} - \ln (x) \frac{d}{d x} [x^{8}]}{x^{16}}$

Step 5.2.2.2

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

$8 \frac{\frac{x \cdot x^{7}}{x \cdot 1} - \ln (x) \frac{d}{d x} [x^{8}]}{x^{16}}$

Step 5.2.2.3

Cancel the common factor.

$8 \frac{\frac{x \cdot x^{7}}{x \cdot 1} - \ln (x) \frac{d}{d x} [x^{8}]}{x^{16}}$

Step 5.2.2.4

Rewrite the expression.

$8 \frac{\frac{x^{7}}{1} - \ln (x) \frac{d}{d x} [x^{8}]}{x^{16}}$

Step 5.2.2.5

Divide $x^{7}$ by $1$ .

$8 \frac{x^{7} - \ln (x) \frac{d}{d x} [x^{8}]}{x^{16}}$

Step 5.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 8$ .

$8 \frac{x^{7} - \ln (x) (8 x^{7})}{x^{16}}$

Step 5.4

Simplify with factoring out.

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Step 5.4.1

Multiply $8$ by $- 1$ .

$8 \frac{x^{7} - 8 \ln (x) x^{7}}{x^{16}}$

Step 5.4.2

Factor $x^{7}$ out of $x^{7} - 8 \ln (x) x^{7}$ .

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Step 5.4.2.1

Multiply by $1$ .

$8 \frac{x^{7} \cdot 1 - 8 \ln (x) x^{7}}{x^{16}}$

Step 5.4.2.2

Factor $x^{7}$ out of $- 8 \ln (x) x^{7}$ .

$8 \frac{x^{7} \cdot 1 + x^{7} (- 8 \ln (x))}{x^{16}}$

Step 5.4.2.3

Factor $x^{7}$ out of $x^{7} \cdot 1 + x^{7} (- 8 \ln (x))$ .

$8 \frac{x^{7} (1 - 8 \ln (x))}{x^{16}}$

Step 6

Cancel the common factors.

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Step 6.1

Factor $x^{7}$ out of $x^{16}$ .

$8 \frac{x^{7} (1 - 8 \ln (x))}{x^{7} x^{9}}$

Step 6.2

Cancel the common factor.

$8 \frac{x^{7} (1 - 8 \ln (x))}{x^{7} x^{9}}$

Step 6.3

Rewrite the expression.

$8 \frac{1 - 8 \ln (x)}{x^{9}}$

Step 7

Combine $8$ and $\frac{1 - 8 \ln (x)}{x^{9}}$ .

$\frac{8 (1 - 8 \ln (x))}{x^{9}}$

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

$\frac{8 \cdot 1 + 8 (- 8 \ln (x))}{x^{9}}$

Step 8.2

Simplify each term.

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Step 8.2.1

Multiply $8$ by $1$ .

$\frac{8 + 8 (- 8 \ln (x))}{x^{9}}$

Step 8.2.2

Simplify $- 8 \ln (x)$ by moving $8$ inside the logarithm.

$\frac{8 + 8 (- \ln (x^{8}))}{x^{9}}$

Step 8.2.3

Multiply $8 (- \ln (x^{8}))$ .

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Step 8.2.3.1

Multiply $- 1$ by $8$ .

$\frac{8 - 8 \ln (x^{8})}{x^{9}}$

Step 8.2.3.2

Simplify $- 8 \ln (x^{8})$ by moving $8$ inside the logarithm.

$\frac{8 - \ln ({(x^{8})}^{8})}{x^{9}}$

Step 8.2.4

Multiply the exponents in ${(x^{8})}^{8}$ .

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Step 8.2.4.1

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

$\frac{8 - \ln (x^{8 \cdot 8})}{x^{9}}$

Step 8.2.4.2

Multiply $8$ by $8$ .

$\frac{8 - \ln (x^{64})}{x^{9}}$