Calculus Examples

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

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 $f (x) = x^{6}$ and $g (x) = 1 + \ln (x)$ .

$\frac{(1 + \ln (x)) \frac{d}{d x} [x^{6}] - x^{6} \frac{d}{d x} [1 + \ln (x)]}{{(1 + \ln (x))}^{2}}$

Step 2

Differentiate.

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Step 2.1

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

$\frac{(1 + \ln (x)) (6 x^{5}) - x^{6} \frac{d}{d x} [1 + \ln (x)]}{{(1 + \ln (x))}^{2}}$

Step 2.2

Move $6$ to the left of $1 + \ln (x)$ .

$\frac{6 \cdot (1 + \ln (x)) x^{5} - x^{6} \frac{d}{d x} [1 + \ln (x)]}{{(1 + \ln (x))}^{2}}$

Step 2.3

By the Sum Rule, the derivative of $1 + \ln (x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [\ln (x)]$ .

$\frac{6 (1 + \ln (x)) x^{5} - x^{6} (\frac{d}{d x} [1] + \frac{d}{d x} [\ln (x)])}{{(1 + \ln (x))}^{2}}$

Step 2.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{6 (1 + \ln (x)) x^{5} - x^{6} (0 + \frac{d}{d x} [\ln (x)])}{{(1 + \ln (x))}^{2}}$

Step 2.5

Add $0$ and $\frac{d}{d x} [\ln (x)]$ .

$\frac{6 (1 + \ln (x)) x^{5} - x^{6} \frac{d}{d x} [\ln (x)]}{{(1 + \ln (x))}^{2}}$

Step 3

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

$\frac{6 (1 + \ln (x)) x^{5} - x^{6} \frac{1}{x}}{{(1 + \ln (x))}^{2}}$

Step 4

Simplify terms.

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Step 4.1

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

$\frac{6 (1 + \ln (x)) x^{5} - \frac{x^{6}}{x}}{{(1 + \ln (x))}^{2}}$

Step 4.2

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

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Step 4.2.1

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

$\frac{6 (1 + \ln (x)) x^{5} - \frac{x \cdot x^{5}}{x}}{{(1 + \ln (x))}^{2}}$

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

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

$\frac{6 (1 + \ln (x)) x^{5} - \frac{x \cdot x^{5}}{x^{1}}}{{(1 + \ln (x))}^{2}}$

Step 4.2.2.2

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

$\frac{6 (1 + \ln (x)) x^{5} - \frac{x \cdot x^{5}}{x \cdot 1}}{{(1 + \ln (x))}^{2}}$

Step 4.2.2.3

Cancel the common factor.

$\frac{6 (1 + \ln (x)) x^{5} - \frac{x \cdot x^{5}}{x \cdot 1}}{{(1 + \ln (x))}^{2}}$

Step 4.2.2.4

Rewrite the expression.

$\frac{6 (1 + \ln (x)) x^{5} - \frac{x^{5}}{1}}{{(1 + \ln (x))}^{2}}$

Step 4.2.2.5

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

$\frac{6 (1 + \ln (x)) x^{5} - x^{5}}{{(1 + \ln (x))}^{2}}$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$\frac{(6 \cdot 1 + 6 \ln (x)) x^{5} - x^{5}}{{(1 + \ln (x))}^{2}}$

Step 5.2

Apply the distributive property.

$\frac{6 \cdot 1 x^{5} + 6 \ln (x) x^{5} - x^{5}}{{(1 + \ln (x))}^{2}}$

Step 5.3

Simplify the numerator.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Multiply $6$ by $1$ .

$\frac{6 x^{5} + 6 \ln (x) x^{5} - x^{5}}{{(1 + \ln (x))}^{2}}$

Step 5.3.1.2

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

$\frac{6 x^{5} + \ln (x^{6}) x^{5} - x^{5}}{{(1 + \ln (x))}^{2}}$

Step 5.3.2

Subtract $x^{5}$ from $6 x^{5}$ .

$\frac{5 x^{5} + \ln (x^{6}) x^{5}}{{(1 + \ln (x))}^{2}}$

Step 5.3.3

Reorder factors in $5 x^{5} + \ln (x^{6}) x^{5}$ .

$\frac{5 x^{5} + x^{5} \ln (x^{6})}{{(1 + \ln (x))}^{2}}$

Step 5.4

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

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Step 5.4.1

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

$\frac{x^{5} \cdot 5 + x^{5} \ln (x^{6})}{{(1 + \ln (x))}^{2}}$

Step 5.4.2

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

$\frac{x^{5} \cdot 5 + x^{5} (\ln (x^{6}))}{{(1 + \ln (x))}^{2}}$

Step 5.4.3

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

$\frac{x^{5} (5 + \ln (x^{6}))}{{(1 + \ln (x))}^{2}}$