Calculus Examples

$- \frac{1}{10} x^{5} - 2 x^{4} - 12 x^{3}$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $- \frac{1}{10} x^{5} - 2 x^{4} - 12 x^{3}$ with respect to $x$ is $\frac{d}{d x} [- \frac{1}{10} x^{5}] + \frac{d}{d x} [- 2 x^{4}] + \frac{d}{d x} [- 12 x^{3}]$ .

$\frac{d}{d x} [- \frac{1}{10} x^{5}] + \frac{d}{d x} [- 2 x^{4}] + \frac{d}{d x} [- 12 x^{3}]$

Step 1.2

Evaluate $\frac{d}{d x} [- \frac{1}{10} x^{5}]$ .

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Step 1.2.1

Since $- \frac{1}{10}$ is constant with respect to $x$ , the derivative of $- \frac{1}{10} x^{5}$ with respect to $x$ is $- \frac{1}{10} \frac{d}{d x} [x^{5}]$ .

$- \frac{1}{10} \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 2 x^{4}] + \frac{d}{d x} [- 12 x^{3}]$

Step 1.2.2

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

$- \frac{1}{10} (5 x^{4}) + \frac{d}{d x} [- 2 x^{4}] + \frac{d}{d x} [- 12 x^{3}]$

Step 1.2.3

Multiply $5$ by $- 1$ .

$- 5 (\frac{1}{10}) x^{4} + \frac{d}{d x} [- 2 x^{4}] + \frac{d}{d x} [- 12 x^{3}]$

Step 1.2.4

Combine $- 5$ and $\frac{1}{10}$ .

$\frac{- 5}{10} x^{4} + \frac{d}{d x} [- 2 x^{4}] + \frac{d}{d x} [- 12 x^{3}]$

Step 1.2.5

Combine $\frac{- 5}{10}$ and $x^{4}$ .

$\frac{- 5 x^{4}}{10} + \frac{d}{d x} [- 2 x^{4}] + \frac{d}{d x} [- 12 x^{3}]$

Step 1.2.6

Cancel the common factor of $- 5$ and $10$ .

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Step 1.2.6.1

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

$\frac{5 (- x^{4})}{10} + \frac{d}{d x} [- 2 x^{4}] + \frac{d}{d x} [- 12 x^{3}]$

Step 1.2.6.2

Cancel the common factors.

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Step 1.2.6.2.1

Factor $5$ out of $10$ .

$\frac{5 (- x^{4})}{5 (2)} + \frac{d}{d x} [- 2 x^{4}] + \frac{d}{d x} [- 12 x^{3}]$

Step 1.2.6.2.2

Cancel the common factor.

$\frac{5 (- x^{4})}{5 \cdot 2} + \frac{d}{d x} [- 2 x^{4}] + \frac{d}{d x} [- 12 x^{3}]$

Step 1.2.6.2.3

Rewrite the expression.

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

Step 1.2.7

Move the negative in front of the fraction.

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

Multiply $4$ by $- 2$ .

$- \frac{x^{4}}{2} - 8 x^{3} + \frac{d}{d x} [- 12 x^{3}]$

Step 1.4

Evaluate $\frac{d}{d x} [- 12 x^{3}]$ .

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Step 1.4.1

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 x^{3}$ with respect to $x$ is $- 12 \frac{d}{d x} [x^{3}]$ .

$- \frac{x^{4}}{2} - 8 x^{3} - 12 \frac{d}{d x} [x^{3}]$

Step 1.4.2

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

$- \frac{x^{4}}{2} - 8 x^{3} - 12 (3 x^{2})$

Step 1.4.3

Multiply $3$ by $- 12$ .

$f' (x) = - \frac{x^{4}}{2} - 8 x^{3} - 36 x^{2}$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $- \frac{x^{4}}{2} - 8 x^{3} - 36 x^{2}$ with respect to $x$ is $\frac{d}{d x} [- \frac{x^{4}}{2}] + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [- 36 x^{2}]$ .

$\frac{d}{d x} [- \frac{x^{4}}{2}] + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [- 36 x^{2}]$

Step 2.2

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

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Step 2.2.1

Since $- \frac{1}{2}$ is constant with respect to $x$ , the derivative of $- \frac{x^{4}}{2}$ with respect to $x$ is $- \frac{1}{2} \frac{d}{d x} [x^{4}]$ .

$- \frac{1}{2} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [- 36 x^{2}]$

Step 2.2.2

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

$- \frac{1}{2} (4 x^{3}) + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [- 36 x^{2}]$

Step 2.2.3

Multiply $4$ by $- 1$ .

$- 4 (\frac{1}{2}) x^{3} + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [- 36 x^{2}]$

Step 2.2.4

Combine $- 4$ and $\frac{1}{2}$ .

$\frac{- 4}{2} x^{3} + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [- 36 x^{2}]$

Step 2.2.5

Combine $\frac{- 4}{2}$ and $x^{3}$ .

$\frac{- 4 x^{3}}{2} + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [- 36 x^{2}]$

Step 2.2.6

Cancel the common factor of $- 4$ and $2$ .

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Step 2.2.6.1

Factor $2$ out of $- 4 x^{3}$ .

$\frac{2 (- 2 x^{3})}{2} + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [- 36 x^{2}]$

Step 2.2.6.2

Cancel the common factors.

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Step 2.2.6.2.1

Factor $2$ out of $2$ .

$\frac{2 (- 2 x^{3})}{2 (1)} + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [- 36 x^{2}]$

Step 2.2.6.2.2

Cancel the common factor.

$\frac{2 (- 2 x^{3})}{2 \cdot 1} + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [- 36 x^{2}]$

Step 2.2.6.2.3

Rewrite the expression.

$\frac{- 2 x^{3}}{1} + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [- 36 x^{2}]$

Step 2.2.6.2.4

Divide $- 2 x^{3}$ by $1$ .

$- 2 x^{3} + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [- 36 x^{2}]$

Step 2.3

Evaluate $\frac{d}{d x} [- 8 x^{3}]$ .

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Step 2.3.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x^{3}$ with respect to $x$ is $- 8 \frac{d}{d x} [x^{3}]$ .

$- 2 x^{3} - 8 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 36 x^{2}]$

Step 2.3.2

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

$- 2 x^{3} - 8 (3 x^{2}) + \frac{d}{d x} [- 36 x^{2}]$

Step 2.3.3

Multiply $3$ by $- 8$ .

$- 2 x^{3} - 24 x^{2} + \frac{d}{d x} [- 36 x^{2}]$

Step 2.4

Evaluate $\frac{d}{d x} [- 36 x^{2}]$ .

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Step 2.4.1

Since $- 36$ is constant with respect to $x$ , the derivative of $- 36 x^{2}$ with respect to $x$ is $- 36 \frac{d}{d x} [x^{2}]$ .

$- 2 x^{3} - 24 x^{2} - 36 \frac{d}{d x} [x^{2}]$

Step 2.4.2

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

$- 2 x^{3} - 24 x^{2} - 36 (2 x)$

Step 2.4.3

Multiply $2$ by $- 36$ .

$f'' (x) = - 2 x^{3} - 24 x^{2} - 72 x$