Calculus Examples

y = 3 x^{2} - 3 x + 1

$y = 3 x^{2} - 3 x + 1$ ,

x = - 4

$x = - 4$

Step 1

By the Sum Rule, the derivative of

3 x^{2} - 3 x + 1

$3 x^{2} - 3 x + 1$ with respect to

x

$x$ is

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

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

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

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

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

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

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

$n = 2$ .

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

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

Step 2.3

Multiply

2

$2$ by

3

$3$ .

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

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

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

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

Step 3

Evaluate

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

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

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Step 3.1

Since

- 3

$- 3$ is constant with respect to

x

$x$ , the derivative of

- 3 x

$- 3 x$ with respect to

x

$x$ is

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

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

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

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

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

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

Step 3.3

Multiply

- 3

$- 3$ by

1

$1$ .

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

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

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

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

Step 4

Differentiate using the Constant Rule.

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Step 4.1

Since

1

$1$ is constant with respect to

x

$x$ , the derivative of

1

$1$ with respect to

x

$x$ is

0

$0$ .

6 x - 3 + 0

$6 x - 3 + 0$

Step 4.2

Add

6 x - 3

$6 x - 3$ and

0

$0$ .

6 x - 3

$6 x - 3$

6 x - 3

$6 x - 3$

Step 5

Evaluate the derivative at

x = - 4

$x = - 4$ .

6 (- 4) - 3

$6 (- 4) - 3$

Step 6

Simplify.

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Step 6.1

Multiply

6

$6$ by

- 4

$- 4$ .

- 24 - 3

$- 24 - 3$

Step 6.2

Subtract

3

$3$ from

- 24

$- 24$ .

- 27

$- 27$

- 27

$- 27$

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