Calculus Examples

$y = - \frac{2}{3} x^{3} + 3 x^{2} + \frac{3}{10} x^{5} - 3$

Step 1

Find the first derivative.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

Multiply $3$ by $- 1$ .

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

Step 1.2.4

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

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

Step 1.2.5

Multiply $- 3$ by $2$ .

$\frac{- 6}{3} x^{2} + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [\frac{3}{10} x^{5}] + \frac{d}{d x} [- 3]$

Step 1.2.6

Combine $\frac{- 6}{3}$ and $x^{2}$ .

$\frac{- 6 x^{2}}{3} + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [\frac{3}{10} x^{5}] + \frac{d}{d x} [- 3]$

Step 1.2.7

Cancel the common factor of $- 6$ and $3$ .

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Step 1.2.7.1

Factor $3$ out of $- 6 x^{2}$ .

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

Step 1.2.7.2

Cancel the common factors.

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Step 1.2.7.2.1

Factor $3$ out of $3$ .

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

Step 1.2.7.2.2

Cancel the common factor.

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

Step 1.2.7.2.3

Rewrite the expression.

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

Step 1.2.7.2.4

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

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

Multiply $2$ by $3$ .

$- 2 x^{2} + 6 x + \frac{d}{d x} [\frac{3}{10} x^{5}] + \frac{d}{d x} [- 3]$

Step 1.4

Evaluate $\frac{d}{d x} [\frac{3}{10} x^{5}]$ .

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Step 1.4.1

Since $\frac{3}{10}$ is constant with respect to $x$ , the derivative of $\frac{3}{10} x^{5}$ with respect to $x$ is $\frac{3}{10} \frac{d}{d x} [x^{5}]$ .

$- 2 x^{2} + 6 x + \frac{3}{10} \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 3]$

Step 1.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$- 2 x^{2} + 6 x + \frac{3}{10} (5 x^{4}) + \frac{d}{d x} [- 3]$

Step 1.4.3

Combine $5$ and $\frac{3}{10}$ .

$- 2 x^{2} + 6 x + \frac{5 \cdot 3}{10} x^{4} + \frac{d}{d x} [- 3]$

Step 1.4.4

Multiply $5$ by $3$ .

$- 2 x^{2} + 6 x + \frac{15}{10} x^{4} + \frac{d}{d x} [- 3]$

Step 1.4.5

Combine $\frac{15}{10}$ and $x^{4}$ .

$- 2 x^{2} + 6 x + \frac{15 x^{4}}{10} + \frac{d}{d x} [- 3]$

Step 1.4.6

Cancel the common factor of $15$ and $10$ .

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Step 1.4.6.1

Factor $5$ out of $15 x^{4}$ .

$- 2 x^{2} + 6 x + \frac{5 (3 x^{4})}{10} + \frac{d}{d x} [- 3]$

Step 1.4.6.2

Cancel the common factors.

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Step 1.4.6.2.1

Factor $5$ out of $10$ .

$- 2 x^{2} + 6 x + \frac{5 (3 x^{4})}{5 (2)} + \frac{d}{d x} [- 3]$

Step 1.4.6.2.2

Cancel the common factor.

$- 2 x^{2} + 6 x + \frac{5 (3 x^{4})}{5 \cdot 2} + \frac{d}{d x} [- 3]$

Step 1.4.6.2.3

Rewrite the expression.

$- 2 x^{2} + 6 x + \frac{3 x^{4}}{2} + \frac{d}{d x} [- 3]$

Step 1.5

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$- 2 x^{2} + 6 x + \frac{3 x^{4}}{2} + 0$

Step 1.6

Simplify.

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Step 1.6.1

Add $- 2 x^{2} + 6 x + \frac{3 x^{4}}{2}$ and $0$ .

$- 2 x^{2} + 6 x + \frac{3 x^{4}}{2}$

Step 1.6.2

Reorder terms.

$f' (x) = \frac{3 x^{4}}{2} - 2 x^{2} + 6 x$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $\frac{3 x^{4}}{2} - 2 x^{2} + 6 x$ with respect to $x$ is $\frac{d}{d x} [\frac{3 x^{4}}{2}] + \frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [6 x]$ .

$\frac{d}{d x} [\frac{3 x^{4}}{2}] + \frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [6 x]$

Step 2.2

Evaluate $\frac{d}{d x} [\frac{3 x^{4}}{2}]$ .

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Step 2.2.1

Since $\frac{3}{2}$ is constant with respect to $x$ , the derivative of $\frac{3 x^{4}}{2}$ with respect to $x$ is $\frac{3}{2} \frac{d}{d x} [x^{4}]$ .

$\frac{3}{2} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [6 x]$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{3}{2} (4 x^{3}) + \frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [6 x]$

Step 2.2.3

Combine $4$ and $\frac{3}{2}$ .

$\frac{4 \cdot 3}{2} x^{3} + \frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [6 x]$

Step 2.2.4

Multiply $4$ by $3$ .

$\frac{12}{2} x^{3} + \frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [6 x]$

Step 2.2.5

Combine $\frac{12}{2}$ and $x^{3}$ .

$\frac{12 x^{3}}{2} + \frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [6 x]$

Step 2.2.6

Cancel the common factor of $12$ and $2$ .

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Step 2.2.6.1

Factor $2$ out of $12 x^{3}$ .

$\frac{2 (6 x^{3})}{2} + \frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [6 x]$

Step 2.2.6.2

Cancel the common factors.

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Step 2.2.6.2.1

Factor $2$ out of $2$ .

$\frac{2 (6 x^{3})}{2 (1)} + \frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [6 x]$

Step 2.2.6.2.2

Cancel the common factor.

$\frac{2 (6 x^{3})}{2 \cdot 1} + \frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [6 x]$

Step 2.2.6.2.3

Rewrite the expression.

$\frac{6 x^{3}}{1} + \frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [6 x]$

Step 2.2.6.2.4

Divide $6 x^{3}$ by $1$ .

$6 x^{3} + \frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [6 x]$

Step 2.3

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

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Step 2.3.1

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

$6 x^{3} - 2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [6 x]$

Step 2.3.2

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

$6 x^{3} - 2 (2 x) + \frac{d}{d x} [6 x]$

Step 2.3.3

Multiply $2$ by $- 2$ .

$6 x^{3} - 4 x + \frac{d}{d x} [6 x]$

Step 2.4

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

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Step 2.4.1

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

$6 x^{3} - 4 x + 6 \frac{d}{d x} [x]$

Step 2.4.2

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

$6 x^{3} - 4 x + 6 \cdot 1$

Step 2.4.3

Multiply $6$ by $1$ .

$f'' (x) = 6 x^{3} - 4 x + 6$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $6 x^{3} - 4 x + 6$ with respect to $x$ is $\frac{d}{d x} [6 x^{3}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [6]$ .

$\frac{d}{d x} [6 x^{3}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [6]$

Step 3.2

Evaluate $\frac{d}{d x} [6 x^{3}]$ .

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Step 3.2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{3}$ with respect to $x$ is $6 \frac{d}{d x} [x^{3}]$ .

$6 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [6]$

Step 3.2.2

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

$6 (3 x^{2}) + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [6]$

Step 3.2.3

Multiply $3$ by $6$ .

$18 x^{2} + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [6]$

Step 3.3

Evaluate $\frac{d}{d x} [- 4 x]$ .

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Step 3.3.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x$ with respect to $x$ is $- 4 \frac{d}{d x} [x]$ .

$18 x^{2} - 4 \frac{d}{d x} [x] + \frac{d}{d x} [6]$

Step 3.3.2

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

$18 x^{2} - 4 \cdot 1 + \frac{d}{d x} [6]$

Step 3.3.3

Multiply $- 4$ by $1$ .

$18 x^{2} - 4 + \frac{d}{d x} [6]$

Step 3.4

Differentiate using the Constant Rule.

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Step 3.4.1

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$18 x^{2} - 4 + 0$

Step 3.4.2

Add $18 x^{2} - 4$ and $0$ .

$f''' (x) = 18 x^{2} - 4$