Calculus Examples

$y = \frac{x^{6}}{12} \cdot \ln (x) - \frac{x^{6}}{72}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{x^{6}}{12} \cdot \ln (x) - \frac{x^{6}}{72})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $\frac{x^{6}}{12} \cdot \ln (x) - \frac{x^{6}}{72}$ with respect to $x$ is $\frac{d}{d x} [\frac{x^{6}}{12} \cdot \ln (x)] + \frac{d}{d x} [- \frac{x^{6}}{72}]$ .

$\frac{d}{d x} [\frac{x^{6}}{12} \cdot \ln (x)] + \frac{d}{d x} [- \frac{x^{6}}{72}]$

Step 3.2

Evaluate $\frac{d}{d x} [\frac{x^{6}}{12} \cdot \ln (x)]$ .

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Step 3.2.1

Combine $\frac{x^{6}}{12}$ and $\ln (x)$ .

$\frac{d}{d x} [\frac{x^{6} \ln (x)}{12}] + \frac{d}{d x} [- \frac{x^{6}}{72}]$

Step 3.2.2

Since $\frac{1}{12}$ is constant with respect to $x$ , the derivative of $\frac{x^{6} \ln (x)}{12}$ with respect to $x$ is $\frac{1}{12} \frac{d}{d x} [x^{6} \ln (x)]$ .

$\frac{1}{12} \frac{d}{d x} [x^{6} \ln (x)] + \frac{d}{d x} [- \frac{x^{6}}{72}]$

Step 3.2.3

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

$\frac{1}{12} (x^{6} \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [x^{6}]) + \frac{d}{d x} [- \frac{x^{6}}{72}]$

Step 3.2.4

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

$\frac{1}{12} (x^{6} \frac{1}{x} + \ln (x) \frac{d}{d x} [x^{6}]) + \frac{d}{d x} [- \frac{x^{6}}{72}]$

Step 3.2.5

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

$\frac{1}{12} (x^{6} \frac{1}{x} + \ln (x) (6 x^{5})) + \frac{d}{d x} [- \frac{x^{6}}{72}]$

Step 3.2.6

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

$\frac{1}{12} (\frac{x^{6}}{x} + \ln (x) (6 x^{5})) + \frac{d}{d x} [- \frac{x^{6}}{72}]$

Step 3.2.7

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

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Step 3.2.7.1

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

$\frac{1}{12} (\frac{x \cdot x^{5}}{x} + \ln (x) (6 x^{5})) + \frac{d}{d x} [- \frac{x^{6}}{72}]$

Step 3.2.7.2

Cancel the common factors.

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Step 3.2.7.2.1

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

$\frac{1}{12} (\frac{x \cdot x^{5}}{x^{1}} + \ln (x) (6 x^{5})) + \frac{d}{d x} [- \frac{x^{6}}{72}]$

Step 3.2.7.2.2

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

$\frac{1}{12} (\frac{x \cdot x^{5}}{x \cdot 1} + \ln (x) (6 x^{5})) + \frac{d}{d x} [- \frac{x^{6}}{72}]$

Step 3.2.7.2.3

Cancel the common factor.

$\frac{1}{12} (\frac{x \cdot x^{5}}{x \cdot 1} + \ln (x) (6 x^{5})) + \frac{d}{d x} [- \frac{x^{6}}{72}]$

Step 3.2.7.2.4

Rewrite the expression.

$\frac{1}{12} (\frac{x^{5}}{1} + \ln (x) (6 x^{5})) + \frac{d}{d x} [- \frac{x^{6}}{72}]$

Step 3.2.7.2.5

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

$\frac{1}{12} (x^{5} + \ln (x) (6 x^{5})) + \frac{d}{d x} [- \frac{x^{6}}{72}]$

Step 3.3

Evaluate $\frac{d}{d x} [- \frac{x^{6}}{72}]$ .

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Step 3.3.1

Since $- \frac{1}{72}$ is constant with respect to $x$ , the derivative of $- \frac{x^{6}}{72}$ with respect to $x$ is $- \frac{1}{72} \frac{d}{d x} [x^{6}]$ .

$\frac{1}{12} (x^{5} + \ln (x) (6 x^{5})) - \frac{1}{72} \frac{d}{d x} [x^{6}]$

Step 3.3.2

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

$\frac{1}{12} (x^{5} + \ln (x) (6 x^{5})) - \frac{1}{72} (6 x^{5})$

Step 3.3.3

Multiply $6$ by $- 1$ .

$\frac{1}{12} (x^{5} + \ln (x) (6 x^{5})) - 6 (\frac{1}{72}) x^{5}$

Step 3.3.4

Combine $- 6$ and $\frac{1}{72}$ .

$\frac{1}{12} (x^{5} + \ln (x) (6 x^{5})) + \frac{- 6}{72} x^{5}$

Step 3.3.5

Combine $\frac{- 6}{72}$ and $x^{5}$ .

$\frac{1}{12} (x^{5} + \ln (x) (6 x^{5})) + \frac{- 6 x^{5}}{72}$

Step 3.3.6

Cancel the common factor of $- 6$ and $72$ .

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Step 3.3.6.1

Factor $6$ out of $- 6 x^{5}$ .

$\frac{1}{12} (x^{5} + \ln (x) (6 x^{5})) + \frac{6 (- x^{5})}{72}$

Step 3.3.6.2

Cancel the common factors.

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Step 3.3.6.2.1

Factor $6$ out of $72$ .

$\frac{1}{12} (x^{5} + \ln (x) (6 x^{5})) + \frac{6 (- x^{5})}{6 (12)}$

Step 3.3.6.2.2

Cancel the common factor.

$\frac{1}{12} (x^{5} + \ln (x) (6 x^{5})) + \frac{6 (- x^{5})}{6 \cdot 12}$

Step 3.3.6.2.3

Rewrite the expression.

$\frac{1}{12} (x^{5} + \ln (x) (6 x^{5})) + \frac{- x^{5}}{12}$

Step 3.3.7

Move the negative in front of the fraction.

$\frac{1}{12} (x^{5} + \ln (x) (6 x^{5})) - \frac{x^{5}}{12}$

Step 3.4

Simplify.

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Step 3.4.1

Apply the distributive property.

$\frac{1}{12} x^{5} + \frac{1}{12} (\ln (x) (6 x^{5})) - \frac{x^{5}}{12}$

Step 3.4.2

Combine terms.

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Step 3.4.2.1

Combine $\ln (x)$ and $\frac{1}{12}$ .

$\frac{1}{12} x^{5} + \frac{\ln (x)}{12} (6 x^{5}) - \frac{x^{5}}{12}$

Step 3.4.2.2

Combine $6$ and $\frac{\ln (x)}{12}$ .

$\frac{1}{12} x^{5} + \frac{6 \ln (x)}{12} x^{5} - \frac{x^{5}}{12}$

Step 3.4.2.3

Combine $\frac{6 \ln (x)}{12}$ and $x^{5}$ .

$\frac{1}{12} x^{5} + \frac{6 \ln (x) x^{5}}{12} - \frac{x^{5}}{12}$

Step 3.4.2.4

Cancel the common factor of $6$ and $12$ .

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Step 3.4.2.4.1

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

$\frac{1}{12} x^{5} + \frac{6 (\ln (x) x^{5})}{12} - \frac{x^{5}}{12}$

Step 3.4.2.4.2

Cancel the common factors.

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Step 3.4.2.4.2.1

Factor $6$ out of $12$ .

$\frac{1}{12} x^{5} + \frac{6 (\ln (x) x^{5})}{6 \cdot 2} - \frac{x^{5}}{12}$

Step 3.4.2.4.2.2

Cancel the common factor.

$\frac{1}{12} x^{5} + \frac{6 (\ln (x) x^{5})}{6 \cdot 2} - \frac{x^{5}}{12}$

Step 3.4.2.4.2.3

Rewrite the expression.

$\frac{1}{12} x^{5} + \frac{\ln (x) x^{5}}{2} - \frac{x^{5}}{12}$

Step 3.4.2.5

To write $\frac{1}{12} x^{5}$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$\frac{\ln (x) x^{5}}{2} + \frac{1}{12} x^{5} \cdot \frac{12}{12} - \frac{x^{5}}{12}$

Step 3.4.2.6

Combine $\frac{1}{12} x^{5}$ and $\frac{12}{12}$ .

$\frac{\ln (x) x^{5}}{2} + \frac{\frac{1}{12} x^{5} \cdot 12}{12} - \frac{x^{5}}{12}$

Step 3.4.2.7

Combine the numerators over the common denominator.

$\frac{\ln (x) x^{5}}{2} + \frac{\frac{1}{12} x^{5} \cdot 12 - x^{5}}{12}$

Step 3.4.2.8

Combine $\frac{1}{12}$ and $x^{5}$ .

$\frac{\ln (x) x^{5}}{2} + \frac{\frac{x^{5}}{12} \cdot 12 - x^{5}}{12}$

Step 3.4.2.9

Combine $\frac{x^{5}}{12}$ and $12$ .

$\frac{\ln (x) x^{5}}{2} + \frac{\frac{x^{5} \cdot 12}{12} - x^{5}}{12}$

Step 3.4.2.10

Move $12$ to the left of $x^{5}$ .

$\frac{\ln (x) x^{5}}{2} + \frac{\frac{12 \cdot x^{5}}{12} - x^{5}}{12}$

Step 3.4.2.11

Cancel the common factor of $12$ .

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Step 3.4.2.11.1

Cancel the common factor.

$\frac{\ln (x) x^{5}}{2} + \frac{\frac{12 x^{5}}{12} - x^{5}}{12}$

Step 3.4.2.11.2

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

$\frac{\ln (x) x^{5}}{2} + \frac{x^{5} - x^{5}}{12}$

Step 3.4.2.12

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

$\frac{\ln (x) x^{5}}{2} + \frac{0}{12}$

Step 3.4.2.13

Cancel the common factor of $0$ and $12$ .

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Step 3.4.2.13.1

Factor $12$ out of $0$ .

$\frac{\ln (x) x^{5}}{2} + \frac{12 (0)}{12}$

Step 3.4.2.13.2

Cancel the common factors.

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Step 3.4.2.13.2.1

Factor $12$ out of $12$ .

$\frac{\ln (x) x^{5}}{2} + \frac{12 \cdot 0}{12 \cdot 1}$

Step 3.4.2.13.2.2

Cancel the common factor.

$\frac{\ln (x) x^{5}}{2} + \frac{12 \cdot 0}{12 \cdot 1}$

Step 3.4.2.13.2.3

Rewrite the expression.

$\frac{\ln (x) x^{5}}{2} + \frac{0}{1}$

Step 3.4.2.13.2.4

Divide $0$ by $1$ .

$\frac{\ln (x) x^{5}}{2} + 0$

Step 3.4.2.14

Add $\frac{\ln (x) x^{5}}{2}$ and $0$ .

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

Step 3.4.3

Reorder terms.

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

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = \frac{x^{5} \ln (x)}{2}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{x^{5} \ln (x)}{2}$