Calculus Examples

$y = \frac{1}{2} x^{2} - x^{4} + x - \frac{1}{4} x^{5}$

Step 1

Find the first derivative.

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Step 1.1

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

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

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

Cancel the common factor of $2$ .

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Step 1.2.5.1

Cancel the common factor.

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

Step 1.2.5.2

Divide $x$ by $1$ .

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

Multiply $4$ by $- 1$ .

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

Step 1.4

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

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

Step 1.5

Evaluate $\frac{d}{d x} [- \frac{1}{4} x^{5}]$ .

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Step 1.5.1

Since $- \frac{1}{4}$ is constant with respect to $x$ , the derivative of $- \frac{1}{4} x^{5}$ with respect to $x$ is $- \frac{1}{4} \frac{d}{d x} [x^{5}]$ .

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

Step 1.5.2

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

$x - 4 x^{3} + 1 - \frac{1}{4} (5 x^{4})$

Step 1.5.3

Multiply $5$ by $- 1$ .

$x - 4 x^{3} + 1 - 5 (\frac{1}{4}) x^{4}$

Step 1.5.4

Combine $- 5$ and $\frac{1}{4}$ .

$x - 4 x^{3} + 1 + \frac{- 5}{4} x^{4}$

Step 1.5.5

Combine $\frac{- 5}{4}$ and $x^{4}$ .

$x - 4 x^{3} + 1 + \frac{- 5 x^{4}}{4}$

Step 1.5.6

Move the negative in front of the fraction.

$x - 4 x^{3} + 1 - \frac{5 x^{4}}{4}$

Step 1.6

Reorder terms.

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

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

Evaluate $\frac{d}{d x} [- \frac{5 x^{4}}{4}]$ .

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Step 2.2.1

Since $- \frac{5}{4}$ is constant with respect to $x$ , the derivative of $- \frac{5 x^{4}}{4}$ with respect to $x$ is $- \frac{5}{4} \frac{d}{d x} [x^{4}]$ .

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $4$ by $- 1$ .

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

Step 2.2.4

Combine $- 4$ and $\frac{5}{4}$ .

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

Step 2.2.5

Multiply $- 4$ by $5$ .

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

Step 2.2.6

Combine $\frac{- 20}{4}$ and $x^{3}$ .

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

Step 2.2.7

Cancel the common factor of $- 20$ and $4$ .

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Step 2.2.7.1

Factor $4$ out of $- 20 x^{3}$ .

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

Step 2.2.7.2

Cancel the common factors.

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Step 2.2.7.2.1

Factor $4$ out of $4$ .

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

Step 2.2.7.2.2

Cancel the common factor.

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

Step 2.2.7.2.3

Rewrite the expression.

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

Step 2.2.7.2.4

Divide $- 5 x^{3}$ by $1$ .

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

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

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

Step 2.3.2

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

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

Step 2.3.3

Multiply $3$ by $- 4$ .

$- 5 x^{3} - 12 x^{2} + \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 2.4

Differentiate.

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Step 2.4.1

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

$- 5 x^{3} - 12 x^{2} + 1 + \frac{d}{d x} [1]$

Step 2.4.2

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

$- 5 x^{3} - 12 x^{2} + 1 + 0$

Step 2.4.3

Add $- 5 x^{3} - 12 x^{2} + 1$ and $0$ .

$f'' (x) = - 5 x^{3} - 12 x^{2} + 1$