Calculus Examples

$- \frac{2}{5} x^{6} + 5 x^{4}$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $- \frac{2}{5} x^{6} + 5 x^{4}$ with respect to $x$ is $\frac{d}{d x} [- \frac{2}{5} x^{6}] + \frac{d}{d x} [5 x^{4}]$ .

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

Step 1.2

Evaluate $\frac{d}{d x} [- \frac{2}{5} x^{6}]$ .

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Step 1.2.1

Since $- \frac{2}{5}$ is constant with respect to $x$ , the derivative of $- \frac{2}{5} x^{6}$ with respect to $x$ is $- \frac{2}{5} \frac{d}{d x} [x^{6}]$ .

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

Step 1.2.2

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

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

Step 1.2.3

Multiply $6$ by $- 1$ .

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

Step 1.2.4

Combine $- 6$ and $\frac{2}{5}$ .

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

Step 1.2.5

Multiply $- 6$ by $2$ .

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

Step 1.2.6

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

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

Step 1.2.7

Move the negative in front of the fraction.

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

Multiply $4$ by $5$ .

$f' (x) = - \frac{12 x^{5}}{5} + 20 x^{3}$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $- \frac{12 x^{5}}{5} + 20 x^{3}$ with respect to $x$ is $\frac{d}{d x} [- \frac{12 x^{5}}{5}] + \frac{d}{d x} [20 x^{3}]$ .

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

Step 2.2

Evaluate $\frac{d}{d x} [- \frac{12 x^{5}}{5}]$ .

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Step 2.2.1

Since $- \frac{12}{5}$ is constant with respect to $x$ , the derivative of $- \frac{12 x^{5}}{5}$ with respect to $x$ is $- \frac{12}{5} \frac{d}{d x} [x^{5}]$ .

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $5$ by $- 1$ .

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

Step 2.2.4

Combine $- 5$ and $\frac{12}{5}$ .

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

Step 2.2.5

Multiply $- 5$ by $12$ .

$\frac{- 60}{5} x^{4} + \frac{d}{d x} [20 x^{3}]$

Step 2.2.6

Combine $\frac{- 60}{5}$ and $x^{4}$ .

$\frac{- 60 x^{4}}{5} + \frac{d}{d x} [20 x^{3}]$

Step 2.2.7

Cancel the common factor of $- 60$ and $5$ .

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Step 2.2.7.1

Factor $5$ out of $- 60 x^{4}$ .

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

Step 2.2.7.2

Cancel the common factors.

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Step 2.2.7.2.1

Factor $5$ out of $5$ .

$\frac{5 (- 12 x^{4})}{5 (1)} + \frac{d}{d x} [20 x^{3}]$

Step 2.2.7.2.2

Cancel the common factor.

$\frac{5 (- 12 x^{4})}{5 \cdot 1} + \frac{d}{d x} [20 x^{3}]$

Step 2.2.7.2.3

Rewrite the expression.

$\frac{- 12 x^{4}}{1} + \frac{d}{d x} [20 x^{3}]$

Step 2.2.7.2.4

Divide $- 12 x^{4}$ by $1$ .

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

$- 12 x^{4} + 20 (3 x^{2})$

Step 2.3.3

Multiply $3$ by $20$ .

$f'' (x) = - 12 x^{4} + 60 x^{2}$