Calculus Examples

$\frac{3 x^{2} - 1}{3}$

Step 1

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{3 x^{2} - 1}{3}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [3 x^{2} - 1]$ .

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Multiply $2$ by $3$ .

$\frac{1}{3} (6 x + \frac{d}{d x} [- 1])$

Step 6

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

$\frac{1}{3} (6 x + 0)$

Step 7

Simplify terms.

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Step 7.1

Add $6 x$ and $0$ .

$\frac{1}{3} (6 x)$

Step 7.2

Combine $6$ and $\frac{1}{3}$ .

$\frac{6}{3} x$

Step 7.3

Combine $\frac{6}{3}$ and $x$ .

$\frac{6 x}{3}$

Step 7.4

Cancel the common factor of $6$ and $3$ .

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Step 7.4.1

Factor $3$ out of $6 x$ .

$\frac{3 (2 x)}{3}$

Step 7.4.2

Cancel the common factors.

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Step 7.4.2.1

Factor $3$ out of $3$ .

$\frac{3 (2 x)}{3 (1)}$

Step 7.4.2.2

Cancel the common factor.

$\frac{3 (2 x)}{3 \cdot 1}$

Step 7.4.2.3

Rewrite the expression.

$\frac{2 x}{1}$

Step 7.4.2.4

Divide $2 x$ by $1$ .

$2 x$