Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

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

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

Step 1.2

Evaluate $\frac{d}{d x} [- \frac{1}{3} x^{4}]$ .

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Step 1.2.1

Since $- \frac{1}{3}$ is constant with respect to $x$ , the derivative of $- \frac{1}{3} x^{4}$ with respect to $x$ is $- \frac{1}{3} \frac{d}{d x} [x^{4}]$ .

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

Step 1.2.2

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

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

Step 1.2.3

Multiply $4$ by $- 1$ .

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

Step 1.2.4

Combine $- 4$ and $\frac{1}{3}$ .

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

Step 1.2.5

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

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

Step 1.2.6

Move the negative in front of the fraction.

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

Multiply $2$ by $2$ .

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

Step 1.4

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

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

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $3$ by $- 1$ .

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

Step 2.2.4

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

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

Step 2.2.5

Multiply $- 3$ by $4$ .

$\frac{- 12}{3} x^{2} + \frac{d}{d x} [4 x] + \frac{d}{d x} [1]$

Step 2.2.6

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

$\frac{- 12 x^{2}}{3} + \frac{d}{d x} [4 x] + \frac{d}{d x} [1]$

Step 2.2.7

Cancel the common factor of $- 12$ and $3$ .

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Step 2.2.7.1

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

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

Step 2.2.7.2

Cancel the common factors.

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Step 2.2.7.2.1

Factor $3$ out of $3$ .

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

Step 2.2.7.2.2

Cancel the common factor.

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

Step 2.2.7.2.3

Rewrite the expression.

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

Step 2.2.7.2.4

Divide $- 4 x^{2}$ by $1$ .

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Multiply $4$ by $1$ .

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

Step 2.4

Differentiate using the Constant Rule.

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Step 2.4.1

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

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

Step 2.4.2

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

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

Step 3

Find the third derivative.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

Multiply $2$ by $- 4$ .

$- 8 x + \frac{d}{d x} [4]$

Step 3.3

Differentiate using the Constant Rule.

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Step 3.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$- 8 x + 0$

Step 3.3.2

Add $- 8 x$ and $0$ .

$f''' (x) = - 8 x$

Step 4

The third derivative of $f (x)$ with respect to $x$ is $- 8 x$ .

$- 8 x$