Calculus Examples

$\frac{2}{15} x^{6} - 3 x^{4}$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $\frac{2}{15} x^{6} - 3 x^{4}$ with respect to $x$ is $\frac{d}{d x} [\frac{2}{15} x^{6}] + \frac{d}{d x} [- 3 x^{4}]$ .

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

Step 1.2

Evaluate $\frac{d}{d x} [\frac{2}{15} x^{6}]$ .

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Step 1.2.1

Since $\frac{2}{15}$ is constant with respect to $x$ , the derivative of $\frac{2}{15} x^{6}$ with respect to $x$ is $\frac{2}{15} \frac{d}{d x} [x^{6}]$ .

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

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$\frac{2}{15} (6 x^{5}) + \frac{d}{d x} [- 3 x^{4}]$

Step 1.2.3

Combine $6$ and $\frac{2}{15}$ .

$\frac{6 \cdot 2}{15} x^{5} + \frac{d}{d x} [- 3 x^{4}]$

Step 1.2.4

Multiply $6$ by $2$ .

$\frac{12}{15} x^{5} + \frac{d}{d x} [- 3 x^{4}]$

Step 1.2.5

Combine $\frac{12}{15}$ and $x^{5}$ .

$\frac{12 x^{5}}{15} + \frac{d}{d x} [- 3 x^{4}]$

Step 1.2.6

Cancel the common factor of $12$ and $15$ .

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Step 1.2.6.1

Factor $3$ out of $12 x^{5}$ .

$\frac{3 (4 x^{5})}{15} + \frac{d}{d x} [- 3 x^{4}]$

Step 1.2.6.2

Cancel the common factors.

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Step 1.2.6.2.1

Factor $3$ out of $15$ .

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

Step 1.2.6.2.2

Cancel the common factor.

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

Step 1.2.6.2.3

Rewrite the expression.

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

Multiply $4$ by $- 3$ .

$f' (x) = \frac{4 x^{5}}{5} - 12 x^{3}$

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Multiply $5$ by $4$ .

$\frac{20}{5} x^{4} + \frac{d}{d x} [- 12 x^{3}]$

Step 2.2.5

Combine $\frac{20}{5}$ and $x^{4}$ .

$\frac{20 x^{4}}{5} + \frac{d}{d x} [- 12 x^{3}]$

Step 2.2.6

Cancel the common factor of $20$ and $5$ .

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Step 2.2.6.1

Factor $5$ out of $20 x^{4}$ .

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

Step 2.2.6.2

Cancel the common factors.

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Step 2.2.6.2.1

Factor $5$ out of $5$ .

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

Step 2.2.6.2.2

Cancel the common factor.

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

Step 2.2.6.2.3

Rewrite the expression.

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

Step 2.2.6.2.4

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Multiply $3$ by $- 12$ .

$f'' (x) = 4 x^{4} - 36 x^{2}$