Calculus Examples

$\frac{x^{3}}{9} \cdot (3 \ln (x - 1))$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Combine $3$ and $\frac{x^{3}}{9}$ .

$\frac{d}{d x} [\frac{3 x^{3}}{9} \cdot \ln (x - 1)]$

Step 1.2

Simplify terms.

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Step 1.2.1

Combine $\frac{3 x^{3}}{9}$ and $\ln (x - 1)$ .

$\frac{d}{d x} [\frac{3 x^{3} \ln (x - 1)}{9}]$

Step 1.2.2

Cancel the common factor of $3$ and $9$ .

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Step 1.2.2.1

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

$\frac{d}{d x} [\frac{3 (x^{3} \ln (x - 1))}{9}]$

Step 1.2.2.2

Cancel the common factors.

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Step 1.2.2.2.1

Factor $3$ out of $9$ .

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

Step 1.2.2.2.2

Cancel the common factor.

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

Step 1.2.2.2.3

Rewrite the expression.

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

Step 1.3

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{x^{3} \ln (x - 1)}{3}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [x^{3} \ln (x - 1)]$ .

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

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{3}$ and $g (x) = \ln (x - 1)$ .

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

Step 3

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 $g (x) = x - 1$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $x - 1$ .

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

Step 3.2

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

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

Step 3.3

Replace all occurrences of $u$ with $x - 1$ .

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

Step 4

Differentiate.

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Step 4.1

Combine $\frac{1}{x - 1}$ and $x^{3}$ .

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

$\frac{1}{3} (\frac{x^{3}}{x - 1} (1 + 0) + \ln (x - 1) \frac{d}{d x} [x^{3}])$

Step 4.5

Simplify the expression.

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Step 4.5.1

Add $1$ and $0$ .

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

Step 4.5.2

Multiply $\frac{x^{3}}{x - 1}$ by $1$ .

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

Step 4.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$\frac{1}{3} (\frac{x^{3}}{x - 1} + \ln (x - 1) (3 x^{2}))$

Step 5

To write $\ln (x - 1) (3 x^{2})$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$\frac{1}{3} (\frac{x^{3}}{x - 1} + \frac{\ln (x - 1) (3 x^{2}) (x - 1)}{x - 1})$

Step 6

Combine the numerators over the common denominator.

$\frac{1}{3} \cdot \frac{x^{3} + \ln (x - 1) (3 x^{2}) (x - 1)}{x - 1}$

Step 7

Multiply $\frac{1}{3}$ by $\frac{x^{3} + \ln (x - 1) (3 x^{2}) (x - 1)}{x - 1}$ .

$\frac{x^{3} + \ln (x - 1) (3 x^{2}) (x - 1)}{3 (x - 1)}$

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

$\frac{x^{3} + \ln (x - 1) (3 x^{2}) (x - 1)}{3 x + 3 \cdot - 1}$

Step 8.2

Simplify the numerator.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{x^{3} + 3 \ln (x - 1) x^{2} (x - 1)}{3 x + 3 \cdot - 1}$

Step 8.2.1.2

Simplify $3 \ln (x - 1)$ by moving $3$ inside the logarithm.

$\frac{x^{3} + \ln ({(x - 1)}^{3}) x^{2} (x - 1)}{3 x + 3 \cdot - 1}$

Step 8.2.1.3

Apply the distributive property.

$\frac{x^{3} + \ln ({(x - 1)}^{3}) x^{2} x + \ln ({(x - 1)}^{3}) x^{2} \cdot - 1}{3 x + 3 \cdot - 1}$

Step 8.2.1.4

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 8.2.1.4.1

Move $x$ .

$\frac{x^{3} + \ln ({(x - 1)}^{3}) (x \cdot x^{2}) + \ln ({(x - 1)}^{3}) x^{2} \cdot - 1}{3 x + 3 \cdot - 1}$

Step 8.2.1.4.2

Multiply $x$ by $x^{2}$ .

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Step 8.2.1.4.2.1

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

$\frac{x^{3} + \ln ({(x - 1)}^{3}) (x^{1} x^{2}) + \ln ({(x - 1)}^{3}) x^{2} \cdot - 1}{3 x + 3 \cdot - 1}$

Step 8.2.1.4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{3} + \ln ({(x - 1)}^{3}) x^{1 + 2} + \ln ({(x - 1)}^{3}) x^{2} \cdot - 1}{3 x + 3 \cdot - 1}$

Step 8.2.1.4.3

Add $1$ and $2$ .

$\frac{x^{3} + \ln ({(x - 1)}^{3}) x^{3} + \ln ({(x - 1)}^{3}) x^{2} \cdot - 1}{3 x + 3 \cdot - 1}$

Step 8.2.1.5

Move $- 1$ to the left of $\ln ({(x - 1)}^{3}) x^{2}$ .

$\frac{x^{3} + \ln ({(x - 1)}^{3}) x^{3} - 1 \cdot (\ln ({(x - 1)}^{3}) x^{2})}{3 x + 3 \cdot - 1}$

Step 8.2.1.6

Rewrite $- 1 (\ln ({(x - 1)}^{3}) x^{2})$ as $- (\ln ({(x - 1)}^{3}) x^{2})$ .

$\frac{x^{3} + \ln ({(x - 1)}^{3}) x^{3} - \ln ({(x - 1)}^{3}) x^{2}}{3 x + 3 \cdot - 1}$

Step 8.2.2

Reorder factors in $x^{3} + \ln ({(x - 1)}^{3}) x^{3} - \ln ({(x - 1)}^{3}) x^{2}$ .

$\frac{x^{3} + x^{3} \ln ({(x - 1)}^{3}) - x^{2} \ln ({(x - 1)}^{3})}{3 x + 3 \cdot - 1}$

Step 8.3

Multiply $3$ by $- 1$ .

$\frac{x^{3} + x^{3} \ln ({(x - 1)}^{3}) - x^{2} \ln ({(x - 1)}^{3})}{3 x - 3}$