Calculus Examples

$f (x) = \frac{10}{3} x^{3} - \frac{51}{2} x^{2} + 5 x + 13$

Step 1

By the Sum Rule, the derivative of $\frac{10}{3} x^{3} - \frac{51}{2} x^{2} + 5 x + 13$ with respect to $x$ is $\frac{d}{d x} [\frac{10}{3} x^{3}] + \frac{d}{d x} [- \frac{51}{2} x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [13]$ .

$\frac{d}{d x} [\frac{10}{3} x^{3}] + \frac{d}{d x} [- \frac{51}{2} x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [13]$

Step 2

Evaluate $\frac{d}{d x} [\frac{10}{3} x^{3}]$ .

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Step 2.1

Since $\frac{10}{3}$ is constant with respect to $x$ , the derivative of $\frac{10}{3} x^{3}$ with respect to $x$ is $\frac{10}{3} \frac{d}{d x} [x^{3}]$ .

$\frac{10}{3} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- \frac{51}{2} x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [13]$

Step 2.2

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

$\frac{10}{3} (3 x^{2}) + \frac{d}{d x} [- \frac{51}{2} x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [13]$

Step 2.3

Combine $3$ and $\frac{10}{3}$ .

$\frac{3 \cdot 10}{3} x^{2} + \frac{d}{d x} [- \frac{51}{2} x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [13]$

Step 2.4

Multiply $3$ by $10$ .

$\frac{30}{3} x^{2} + \frac{d}{d x} [- \frac{51}{2} x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [13]$

Step 2.5

Combine $\frac{30}{3}$ and $x^{2}$ .

$\frac{30 x^{2}}{3} + \frac{d}{d x} [- \frac{51}{2} x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [13]$

Step 2.6

Cancel the common factor of $30$ and $3$ .

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Step 2.6.1

Factor $3$ out of $30 x^{2}$ .

$\frac{3 (10 x^{2})}{3} + \frac{d}{d x} [- \frac{51}{2} x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [13]$

Step 2.6.2

Cancel the common factors.

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Step 2.6.2.1

Factor $3$ out of $3$ .

$\frac{3 (10 x^{2})}{3 (1)} + \frac{d}{d x} [- \frac{51}{2} x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [13]$

Step 2.6.2.2

Cancel the common factor.

$\frac{3 (10 x^{2})}{3 \cdot 1} + \frac{d}{d x} [- \frac{51}{2} x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [13]$

Step 2.6.2.3

Rewrite the expression.

$\frac{10 x^{2}}{1} + \frac{d}{d x} [- \frac{51}{2} x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [13]$

Step 2.6.2.4

Divide $10 x^{2}$ by $1$ .

$10 x^{2} + \frac{d}{d x} [- \frac{51}{2} x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [13]$

Step 3

Evaluate $\frac{d}{d x} [- \frac{51}{2} x^{2}]$ .

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Step 3.1

Since $- \frac{51}{2}$ is constant with respect to $x$ , the derivative of $- \frac{51}{2} x^{2}$ with respect to $x$ is $- \frac{51}{2} \frac{d}{d x} [x^{2}]$ .

$10 x^{2} - \frac{51}{2} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [13]$

Step 3.2

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

$10 x^{2} - \frac{51}{2} (2 x) + \frac{d}{d x} [5 x] + \frac{d}{d x} [13]$

Step 3.3

Multiply $2$ by $- 1$ .

$10 x^{2} - 2 (\frac{51}{2}) x + \frac{d}{d x} [5 x] + \frac{d}{d x} [13]$

Step 3.4

Combine $- 2$ and $\frac{51}{2}$ .

$10 x^{2} + \frac{- 2 \cdot 51}{2} x + \frac{d}{d x} [5 x] + \frac{d}{d x} [13]$

Step 3.5

Multiply $- 2$ by $51$ .

$10 x^{2} + \frac{- 102}{2} x + \frac{d}{d x} [5 x] + \frac{d}{d x} [13]$

Step 3.6

Combine $\frac{- 102}{2}$ and $x$ .

$10 x^{2} + \frac{- 102 x}{2} + \frac{d}{d x} [5 x] + \frac{d}{d x} [13]$

Step 3.7

Cancel the common factor of $- 102$ and $2$ .

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Step 3.7.1

Factor $2$ out of $- 102 x$ .

$10 x^{2} + \frac{2 (- 51 x)}{2} + \frac{d}{d x} [5 x] + \frac{d}{d x} [13]$

Step 3.7.2

Cancel the common factors.

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Step 3.7.2.1

Factor $2$ out of $2$ .

$10 x^{2} + \frac{2 (- 51 x)}{2 (1)} + \frac{d}{d x} [5 x] + \frac{d}{d x} [13]$

Step 3.7.2.2

Cancel the common factor.

$10 x^{2} + \frac{2 (- 51 x)}{2 \cdot 1} + \frac{d}{d x} [5 x] + \frac{d}{d x} [13]$

Step 3.7.2.3

Rewrite the expression.

$10 x^{2} + \frac{- 51 x}{1} + \frac{d}{d x} [5 x] + \frac{d}{d x} [13]$

Step 3.7.2.4

Divide $- 51 x$ by $1$ .

$10 x^{2} - 51 x + \frac{d}{d x} [5 x] + \frac{d}{d x} [13]$

Step 4

Evaluate $\frac{d}{d x} [5 x]$ .

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Step 4.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$10 x^{2} - 51 x + 5 \frac{d}{d x} [x] + \frac{d}{d x} [13]$

Step 4.2

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

$10 x^{2} - 51 x + 5 \cdot 1 + \frac{d}{d x} [13]$

Step 4.3

Multiply $5$ by $1$ .

$10 x^{2} - 51 x + 5 + \frac{d}{d x} [13]$

Step 5

Differentiate using the Constant Rule.

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Step 5.1

Since $13$ is constant with respect to $x$ , the derivative of $13$ with respect to $x$ is $0$ .

$10 x^{2} - 51 x + 5 + 0$

Step 5.2

Add $10 x^{2} - 51 x + 5$ and $0$ .

$10 x^{2} - 51 x + 5$