Calculus Examples

$f (x) = - \frac{1}{2} x^{2} + x^{3} + x^{4} - \frac{3}{20} x^{5}$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $- \frac{1}{2} x^{2} + x^{3} + x^{4} - \frac{3}{20} x^{5}$ with respect to $x$ is $\frac{d}{d x} [- \frac{1}{2} x^{2}] + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- \frac{3}{20} x^{5}]$ .

$\frac{d}{d x} [- \frac{1}{2} x^{2}] + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- \frac{3}{20} x^{5}]$

Step 1.2

Evaluate $\frac{d}{d x} [- \frac{1}{2} x^{2}]$ .

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Step 1.2.1

Since $- \frac{1}{2}$ is constant with respect to $x$ , the derivative of $- \frac{1}{2} x^{2}$ with respect to $x$ is $- \frac{1}{2} \frac{d}{d x} [x^{2}]$ .

$- \frac{1}{2} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- \frac{3}{20} x^{5}]$

Step 1.2.2

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

$- \frac{1}{2} (2 x) + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- \frac{3}{20} x^{5}]$

Step 1.2.3

Multiply $2$ by $- 1$ .

$- 2 (\frac{1}{2}) x + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- \frac{3}{20} x^{5}]$

Step 1.2.4

Combine $- 2$ and $\frac{1}{2}$ .

$\frac{- 2}{2} x + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- \frac{3}{20} x^{5}]$

Step 1.2.5

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

$\frac{- 2 x}{2} + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- \frac{3}{20} x^{5}]$

Step 1.2.6

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

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Step 1.2.6.1

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

$\frac{2 (- x)}{2} + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- \frac{3}{20} x^{5}]$

Step 1.2.6.2

Cancel the common factors.

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Step 1.2.6.2.1

Factor $2$ out of $2$ .

$\frac{2 (- x)}{2 (1)} + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- \frac{3}{20} x^{5}]$

Step 1.2.6.2.2

Cancel the common factor.

$\frac{2 (- x)}{2 \cdot 1} + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- \frac{3}{20} x^{5}]$

Step 1.2.6.2.3

Rewrite the expression.

$\frac{- x}{1} + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- \frac{3}{20} x^{5}]$

Step 1.2.6.2.4

Divide $- x$ by $1$ .

$- x + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- \frac{3}{20} x^{5}]$

Step 1.3

Differentiate using the Power Rule.

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Step 1.3.1

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

$- x + 3 x^{2} + \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- \frac{3}{20} x^{5}]$

Step 1.3.2

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

$- x + 3 x^{2} + 4 x^{3} + \frac{d}{d x} [- \frac{3}{20} x^{5}]$

Step 1.4

Evaluate $\frac{d}{d x} [- \frac{3}{20} x^{5}]$ .

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Step 1.4.1

Since $- \frac{3}{20}$ is constant with respect to $x$ , the derivative of $- \frac{3}{20} x^{5}$ with respect to $x$ is $- \frac{3}{20} \frac{d}{d x} [x^{5}]$ .

$- x + 3 x^{2} + 4 x^{3} - \frac{3}{20} \frac{d}{d x} [x^{5}]$

Step 1.4.2

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

$- x + 3 x^{2} + 4 x^{3} - \frac{3}{20} (5 x^{4})$

Step 1.4.3

Multiply $5$ by $- 1$ .

$- x + 3 x^{2} + 4 x^{3} - 5 (\frac{3}{20}) x^{4}$

Step 1.4.4

Combine $- 5$ and $\frac{3}{20}$ .

$- x + 3 x^{2} + 4 x^{3} + \frac{- 5 \cdot 3}{20} x^{4}$

Step 1.4.5

Multiply $- 5$ by $3$ .

$- x + 3 x^{2} + 4 x^{3} + \frac{- 15}{20} x^{4}$

Step 1.4.6

Combine $\frac{- 15}{20}$ and $x^{4}$ .

$- x + 3 x^{2} + 4 x^{3} + \frac{- 15 x^{4}}{20}$

Step 1.4.7

Cancel the common factor of $- 15$ and $20$ .

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Step 1.4.7.1

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

$- x + 3 x^{2} + 4 x^{3} + \frac{5 (- 3 x^{4})}{20}$

Step 1.4.7.2

Cancel the common factors.

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Step 1.4.7.2.1

Factor $5$ out of $20$ .

$- x + 3 x^{2} + 4 x^{3} + \frac{5 (- 3 x^{4})}{5 (4)}$

Step 1.4.7.2.2

Cancel the common factor.

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

Step 1.4.7.2.3

Rewrite the expression.

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

Step 1.4.8

Move the negative in front of the fraction.

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

Step 1.5

Reorder terms.

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

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $4$ by $- 1$ .

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

Step 2.2.4

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

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

Step 2.2.5

Multiply $- 4$ by $3$ .

$\frac{- 12}{4} x^{3} + \frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- x]$

Step 2.2.6

Combine $\frac{- 12}{4}$ and $x^{3}$ .

$\frac{- 12 x^{3}}{4} + \frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- x]$

Step 2.2.7

Cancel the common factor of $- 12$ and $4$ .

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Step 2.2.7.1

Factor $4$ out of $- 12 x^{3}$ .

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

Step 2.2.7.2

Cancel the common factors.

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Step 2.2.7.2.1

Factor $4$ out of $4$ .

$\frac{4 (- 3 x^{3})}{4 (1)} + \frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- x]$

Step 2.2.7.2.2

Cancel the common factor.

$\frac{4 (- 3 x^{3})}{4 \cdot 1} + \frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- x]$

Step 2.2.7.2.3

Rewrite the expression.

$\frac{- 3 x^{3}}{1} + \frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- x]$

Step 2.2.7.2.4

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Multiply $3$ by $4$ .

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

Multiply $2$ by $3$ .

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

Step 2.5

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

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Step 2.5.1

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

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

Step 2.5.2

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

$- 3 x^{3} + 12 x^{2} + 6 x - 1 \cdot 1$

Step 2.5.3

Multiply $- 1$ by $1$ .

$f'' (x) = - 3 x^{3} + 12 x^{2} + 6 x - 1$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $- 3 x^{3} + 12 x^{2} + 6 x - 1$ .

$- 3 x^{3} + 12 x^{2} + 6 x - 1$