Calculus Examples

$y = \frac{12}{18} x^{18} - 15 x^{2}$

Step 1

Differentiate using the Sum Rule.

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Step 1.1

Cancel the common factor of $12$ and $18$ .

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Step 1.1.1

Factor $6$ out of $12$ .

$\frac{d}{d x} [\frac{6 (2)}{18} x^{18} - 15 x^{2}]$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Factor $6$ out of $18$ .

$\frac{d}{d x} [\frac{6 \cdot 2}{6 \cdot 3} x^{18} - 15 x^{2}]$

Step 1.1.2.2

Cancel the common factor.

$\frac{d}{d x} [\frac{6 \cdot 2}{6 \cdot 3} x^{18} - 15 x^{2}]$

Step 1.1.2.3

Rewrite the expression.

$\frac{d}{d x} [\frac{2}{3} x^{18} - 15 x^{2}]$

Step 1.2

By the Sum Rule, the derivative of $\frac{2}{3} x^{18} - 15 x^{2}$ with respect to $x$ is $\frac{d}{d x} [\frac{2}{3} x^{18}] + \frac{d}{d x} [- 15 x^{2}]$ .

$\frac{d}{d x} [\frac{2}{3} x^{18}] + \frac{d}{d x} [- 15 x^{2}]$

Step 2

Evaluate $\frac{d}{d x} [\frac{2}{3} x^{18}]$ .

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Step 2.1

Since $\frac{2}{3}$ is constant with respect to $x$ , the derivative of $\frac{2}{3} x^{18}$ with respect to $x$ is $\frac{2}{3} \frac{d}{d x} [x^{18}]$ .

$\frac{2}{3} \frac{d}{d x} [x^{18}] + \frac{d}{d x} [- 15 x^{2}]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 18$ .

$\frac{2}{3} (18 x^{17}) + \frac{d}{d x} [- 15 x^{2}]$

Step 2.3

Combine $18$ and $\frac{2}{3}$ .

$\frac{18 \cdot 2}{3} x^{17} + \frac{d}{d x} [- 15 x^{2}]$

Step 2.4

Multiply $18$ by $2$ .

$\frac{36}{3} x^{17} + \frac{d}{d x} [- 15 x^{2}]$

Step 2.5

Combine $\frac{36}{3}$ and $x^{17}$ .

$\frac{36 x^{17}}{3} + \frac{d}{d x} [- 15 x^{2}]$

Step 2.6

Cancel the common factor of $36$ and $3$ .

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Step 2.6.1

Factor $3$ out of $36 x^{17}$ .

$\frac{3 (12 x^{17})}{3} + \frac{d}{d x} [- 15 x^{2}]$

Step 2.6.2

Cancel the common factors.

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Step 2.6.2.1

Factor $3$ out of $3$ .

$\frac{3 (12 x^{17})}{3 (1)} + \frac{d}{d x} [- 15 x^{2}]$

Step 2.6.2.2

Cancel the common factor.

$\frac{3 (12 x^{17})}{3 \cdot 1} + \frac{d}{d x} [- 15 x^{2}]$

Step 2.6.2.3

Rewrite the expression.

$\frac{12 x^{17}}{1} + \frac{d}{d x} [- 15 x^{2}]$

Step 2.6.2.4

Divide $12 x^{17}$ by $1$ .

$12 x^{17} + \frac{d}{d x} [- 15 x^{2}]$

Step 3

Evaluate $\frac{d}{d x} [- 15 x^{2}]$ .

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Step 3.1

Since $- 15$ is constant with respect to $x$ , the derivative of $- 15 x^{2}$ with respect to $x$ is $- 15 \frac{d}{d x} [x^{2}]$ .

$12 x^{17} - 15 \frac{d}{d x} [x^{2}]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$12 x^{17} - 15 (2 x)$

Step 3.3

Multiply $2$ by $- 15$ .

$12 x^{17} - 30 x$