Calculus Examples

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

Step 1

By the Sum Rule, the derivative of $\frac{x^{3}}{3} \cdot \ln (x) - \frac{x^{3}}{9}$ with respect to $x$ is $\frac{d}{d x} [\frac{x^{3}}{3} \cdot \ln (x)] + \frac{d}{d x} [- \frac{x^{3}}{9}]$ .

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

Step 2

Evaluate $\frac{d}{d x} [\frac{x^{3}}{3} \cdot \ln (x)]$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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 $g (x) = \ln (x)$ .

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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Step 2.7.1

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

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

Step 2.7.2

Cancel the common factors.

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Step 2.7.2.1

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

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

Step 2.7.2.2

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

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

Step 2.7.2.3

Cancel the common factor.

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

Step 2.7.2.4

Rewrite the expression.

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

Step 2.7.2.5

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

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

Step 3

Evaluate $\frac{d}{d x} [- \frac{x^{3}}{9}]$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Multiply $3$ by $- 1$ .

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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Step 3.6.1

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

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

Step 3.6.2

Cancel the common factors.

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Step 3.6.2.1

Factor $3$ out of $9$ .

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

Step 3.6.2.2

Cancel the common factor.

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

Step 3.6.2.3

Rewrite the expression.

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

Step 3.7

Move the negative in front of the fraction.

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Combine terms.

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Step 4.2.1

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

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

Step 4.2.2

Combine $3$ and $\frac{\ln (x)}{3}$ .

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

Step 4.2.3

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

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

Step 4.2.4

Cancel the common factor of $3$ .

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Step 4.2.4.1

Cancel the common factor.

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

Step 4.2.4.2

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

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

Step 4.2.5

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

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

Step 4.2.6

Combine $\frac{1}{3} x^{2}$ and $\frac{3}{3}$ .

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

Step 4.2.7

Combine the numerators over the common denominator.

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

Step 4.2.8

Combine $\frac{1}{3}$ and $x^{2}$ .

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

Step 4.2.9

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

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

Step 4.2.10

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

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

Step 4.2.11

Cancel the common factor of $3$ .

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Step 4.2.11.1

Cancel the common factor.

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

Step 4.2.11.2

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

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

Step 4.2.12

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

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

Step 4.2.13

Cancel the common factor of $0$ and $3$ .

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Step 4.2.13.1

Factor $3$ out of $0$ .

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

Step 4.2.13.2

Cancel the common factors.

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Step 4.2.13.2.1

Factor $3$ out of $3$ .

$\ln (x) x^{2} + \frac{3 \cdot 0}{3 \cdot 1}$

Step 4.2.13.2.2

Cancel the common factor.

$\ln (x) x^{2} + \frac{3 \cdot 0}{3 \cdot 1}$

Step 4.2.13.2.3

Rewrite the expression.

$\ln (x) x^{2} + \frac{0}{1}$

Step 4.2.13.2.4

Divide $0$ by $1$ .

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

Step 4.2.14

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

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

Step 4.3

Reorder the factors of $\ln (x) x^{2}$ .

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