Calculus Examples

3 x^{2} - x

$3 x^{2} - x$

Step 1

By the Sum Rule, the derivative of

3 x^{2} - x

$3 x^{2} - x$ with respect to

x

$x$ is

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

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

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

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

Step 2

Evaluate

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

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

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Step 2.1

Since

3

$3$ is constant with respect to

x

$x$ , the derivative of

3 x^{2}

$3 x^{2}$ with respect to

x

$x$ is

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

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

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

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

Step 2.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 2

$n = 2$ .

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

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

Step 2.3

Multiply

2

$2$ by

3

$3$ .

6 x + \frac{d}{d x} [- x]

$6 x + \frac{d}{d x} [- x]$

6 x + \frac{d}{d x} [- x]

$6 x + \frac{d}{d x} [- x]$

Step 3

Evaluate

\frac{d}{d x} [- x]

$\frac{d}{d x} [- x]$ .

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Step 3.1

Since

- 1

$- 1$ is constant with respect to

x

$x$ , the derivative of

- x

$- x$ with respect to

x

$x$ is

- \frac{d}{d x} [x]

$- \frac{d}{d x} [x]$ .

6 x - \frac{d}{d x} [x]

$6 x - \frac{d}{d x} [x]$

Step 3.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

6 x - 1 \cdot 1

$6 x - 1 \cdot 1$

Step 3.3

Multiply

- 1

$- 1$ by

1

$1$ .

6 x - 1

$6 x - 1$

6 x - 1

$6 x - 1$