Calculus Examples

2 x^{2} - 5 x + 3

$2 x^{2} - 5 x + 3$

Step 1

By the Sum Rule, the derivative of

2 x^{2} - 5 x + 3

$2 x^{2} - 5 x + 3$ with respect to

x

$x$ is

\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [3]

$\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [3]$ .

\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [3]

$\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [3]$

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} [- 5 x] + \frac{d}{d x} [3]

$2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [3]$

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} [- 5 x] + \frac{d}{d x} [3]

$2 (2 x) + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [3]$

Step 2.3

Multiply

2

$2$ by

2

$2$ .

4 x + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [3]

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

4 x + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [3]

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

Step 3

Evaluate

\frac{d}{d x} [- 5 x]

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

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Step 3.1

Since

- 5

$- 5$ is constant with respect to

x

$x$ , the derivative of

- 5 x

$- 5 x$ with respect to

x

$x$ is

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

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

4 x - 5 \frac{d}{d x} [x] + \frac{d}{d x} [3]

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

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

$n = 1$ .

4 x - 5 \cdot 1 + \frac{d}{d x} [3]

$4 x - 5 \cdot 1 + \frac{d}{d x} [3]$

Step 3.3

Multiply

- 5

$- 5$ by

1

$1$ .

4 x - 5 + \frac{d}{d x} [3]

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

4 x - 5 + \frac{d}{d x} [3]

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

Step 4

Differentiate using the Constant Rule.

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Step 4.1

Since

3

$3$ is constant with respect to

x

$x$ , the derivative of

3

$3$ with respect to

x

$x$ is

0

$0$ .

4 x - 5 + 0

$4 x - 5 + 0$

Step 4.2

Add

4 x - 5

$4 x - 5$ and

0

$0$ .

4 x - 5

$4 x - 5$

4 x - 5

$4 x - 5$