Calculus Examples

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

Step 1

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

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

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $3 \ln (x) - 1$ with respect to $x$ is $\frac{d}{d x} [3 \ln (x)] + \frac{d}{d x} [- 1]$ .

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

Step 2.2

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

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

Step 3

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

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

Step 4

Differentiate.

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Step 4.1

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

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

Step 4.2

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

$\frac{x^{3}}{9} (\frac{3}{x} + 0) + (3 \ln (x) - 1) \frac{d}{d x} [\frac{x^{3}}{9}]$

Step 4.3

Simplify terms.

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Step 4.3.1

Add $\frac{3}{x}$ and $0$ .

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

Step 4.3.2

Multiply $\frac{x^{3}}{9}$ by $\frac{3}{x}$ .

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

Step 4.3.3

Move $3$ to the left of $x^{3}$ .

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

Step 4.3.4

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

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Step 4.3.4.1

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

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

Step 4.3.4.2

Cancel the common factors.

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Step 4.3.4.2.1

Factor $3$ out of $9 x$ .

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

Step 4.3.4.2.2

Cancel the common factor.

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

Step 4.3.4.2.3

Rewrite the expression.

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

Step 4.3.5

Cancel the common factor of $x^{3}$ and $x$ .

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Step 4.3.5.1

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

$\frac{x \cdot x^{2}}{3 x} + (3 \ln (x) - 1) \frac{d}{d x} [\frac{x^{3}}{9}]$

Step 4.3.5.2

Cancel the common factors.

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Step 4.3.5.2.1

Factor $x$ out of $3 x$ .

$\frac{x \cdot x^{2}}{x \cdot 3} + (3 \ln (x) - 1) \frac{d}{d x} [\frac{x^{3}}{9}]$

Step 4.3.5.2.2

Cancel the common factor.

$\frac{x \cdot x^{2}}{x \cdot 3} + (3 \ln (x) - 1) \frac{d}{d x} [\frac{x^{3}}{9}]$

Step 4.3.5.2.3

Rewrite the expression.

$\frac{x^{2}}{3} + (3 \ln (x) - 1) \frac{d}{d x} [\frac{x^{3}}{9}]$

Step 4.4

Since $\frac{1}{9}$ is constant with respect to $x$ , the derivative of $\frac{x^{3}}{9}$ with respect to $x$ is $\frac{1}{9} \frac{d}{d x} [x^{3}]$ .

$\frac{x^{2}}{3} + (3 \ln (x) - 1) (\frac{1}{9} \frac{d}{d x} [x^{3}])$

Step 4.5

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

$\frac{x^{2}}{3} + (3 \ln (x) - 1) \frac{1}{9} (3 x^{2})$

Step 4.6

Simplify terms.

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Step 4.6.1

Combine $3$ and $\frac{1}{9}$ .

$\frac{x^{2}}{3} + (3 \ln (x) - 1) \frac{3}{9} x^{2}$

Step 4.6.2

Combine $x^{2}$ and $\frac{3}{9}$ .

$\frac{x^{2}}{3} + (3 \ln (x) - 1) \frac{x^{2} \cdot 3}{9}$

Step 4.6.3

Move $3$ to the left of $x^{2}$ .

$\frac{x^{2}}{3} + (3 \ln (x) - 1) \frac{3 \cdot x^{2}}{9}$

Step 4.6.4

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

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Step 4.6.4.1

Factor $3$ out of $3 x^{2}$ .

$\frac{x^{2}}{3} + (3 \ln (x) - 1) \frac{3 (x^{2})}{9}$

Step 4.6.4.2

Cancel the common factors.

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Step 4.6.4.2.1

Factor $3$ out of $9$ .

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

Step 4.6.4.2.2

Cancel the common factor.

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

Step 4.6.4.2.3

Rewrite the expression.

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

Step 5

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

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

Step 6

Combine $(3 \ln (x) - 1) \frac{x^{2}}{3}$ and $\frac{3}{3}$ .

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

Step 7

Combine the numerators over the common denominator.

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

Step 8

Combine $3$ and $\frac{x^{2}}{3}$ .

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

Step 9

Cancel the common factor of $3$ .

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Step 9.1

Cancel the common factor.

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

Step 9.2

Divide $x^{2}$ by $1$ .

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

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

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

Step 10.2

Simplify the numerator.

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Step 10.2.1

Simplify each term.

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Step 10.2.1.1

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

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

Step 10.2.1.2

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

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

Step 10.2.2

Combine the opposite terms in $x^{2} + \ln (x^{3}) x^{2} - x^{2}$ .

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Step 10.2.2.1

Subtract $x^{2}$ from $x^{2}$ .

$\frac{\ln (x^{3}) x^{2} + 0}{3}$

Step 10.2.2.2

Add $\ln (x^{3}) x^{2}$ and $0$ .

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

Step 10.2.3

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

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

Step 10.3

Expand $\ln (x^{3})$ by moving $3$ outside the logarithm.

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

Step 10.4

Cancel the common factor of $3$ .

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Step 10.4.1

Cancel the common factor.

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

Step 10.4.2

Divide $x^{2} (\ln (x))$ by $1$ .

$x^{2} (\ln (x))$