Calculus Examples

$\frac{1}{2} x^{4} + 2 x^{3}$

Step 1

Find the first derivative.

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Step 1.1

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

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

Step 1.2

Evaluate $\frac{d}{d x} [\frac{1}{2} x^{4}]$ .

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

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

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Step 1.2.5.1

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

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

Step 1.2.5.2

Cancel the common factors.

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Step 1.2.5.2.1

Factor $2$ out of $2$ .

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

Step 1.2.5.2.2

Cancel the common factor.

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

Step 1.2.5.2.3

Rewrite the expression.

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

Step 1.2.5.2.4

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

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

Step 1.3

Evaluate $\frac{d}{d x} [2 x^{3}]$ .

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Step 1.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{3}$ with respect to $x$ is $2 \frac{d}{d x} [x^{3}]$ .

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

Step 1.3.2

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

$2 x^{3} + 2 (3 x^{2})$

Step 1.3.3

Multiply $3$ by $2$ .

$f' (x) = 2 x^{3} + 6 x^{2}$

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

Evaluate $\frac{d}{d x} [2 x^{3}]$ .

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Step 2.2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{3}$ with respect to $x$ is $2 \frac{d}{d x} [x^{3}]$ .

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $3$ by $2$ .

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

$6 x^{2} + 6 (2 x)$

Step 2.3.3

Multiply $2$ by $6$ .

$f'' (x) = 6 x^{2} + 12 x$