Calculus Examples

2 x^{2} + x - 1

$2 x^{2} + x - 1$

Step 1

By the Sum Rule, the derivative of

2 x^{2} + x - 1

$2 x^{2} + x - 1$ with respect to

x

$x$ is

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

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

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

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

Step 2

Evaluate

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

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

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Step 2.1

Since

2

$2$ is constant with respect to

x

$x$ , the derivative of

2 x^{2}

$2 x^{2}$ with respect to

x

$x$ is

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

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

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

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

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

$n = 2$ .

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

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

Step 2.3

Multiply

2

$2$ by

2

$2$ .

4 x + \frac{d}{d x} [x] + \frac{d}{d x} [- 1]

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

4 x + \frac{d}{d x} [x] + \frac{d}{d x} [- 1]

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

Step 3

Differentiate.

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Step 3.1

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

$n = 1$ .

4 x + 1 + \frac{d}{d x} [- 1]

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

Step 3.2

Since

- 1

$- 1$ is constant with respect to

x

$x$ , the derivative of

- 1

$- 1$ with respect to

x

$x$ is

0

$0$ .

4 x + 1 + 0

$4 x + 1 + 0$

Step 3.3

Add

4 x + 1

$4 x + 1$ and

0

$0$ .

4 x + 1

$4 x + 1$

4 x + 1

$4 x + 1$