Calculus Examples

f (x) = x^{2} + 3 x + 34

$f (x) = x^{2} + 3 x + 34$ ,

(- 3, 4)

$(- 3, 4)$

Step 1

Find the derivative.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1

Differentiate.

Tap for more steps...

Step 1.1.1.1

By the Sum Rule, the derivative of

x^{2} + 3 x + 34

$x^{2} + 3 x + 34$ with respect to

x

$x$ is

\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [34]

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [34]$ .

\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [34]

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [34]$

Step 1.1.1.2

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 x + \frac{d}{d x} [3 x] + \frac{d}{d x} [34]

$2 x + \frac{d}{d x} [3 x] + \frac{d}{d x} [34]$

2 x + \frac{d}{d x} [3 x] + \frac{d}{d x} [34]

$2 x + \frac{d}{d x} [3 x] + \frac{d}{d x} [34]$

Step 1.1.2

Evaluate

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

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

Tap for more steps...

Step 1.1.2.1

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]

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

2 x + 3 \frac{d}{d x} [x] + \frac{d}{d x} [34]

$2 x + 3 \frac{d}{d x} [x] + \frac{d}{d x} [34]$

Step 1.1.2.2

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 = 1

$n = 1$ .

2 x + 3 \cdot 1 + \frac{d}{d x} [34]

$2 x + 3 \cdot 1 + \frac{d}{d x} [34]$

Step 1.1.2.3

Multiply

3

$3$ by

1

$1$ .

2 x + 3 + \frac{d}{d x} [34]

$2 x + 3 + \frac{d}{d x} [34]$

2 x + 3 + \frac{d}{d x} [34]

$2 x + 3 + \frac{d}{d x} [34]$

Step 1.1.3

Differentiate using the Constant Rule.

Tap for more steps...

Step 1.1.3.1

Since

34

$34$ is constant with respect to

x

$x$ , the derivative of

34

$34$ with respect to

x

$x$ is

0

$0$ .

2 x + 3 + 0

$2 x + 3 + 0$

Step 1.1.3.2

Add

2 x + 3

$2 x + 3$ and

0

$0$ .

f' (x) = 2 x + 3

$f' (x) = 2 x + 3$

f' (x) = 2 x + 3

$f' (x) = 2 x + 3$

f' (x) = 2 x + 3

$f' (x) = 2 x + 3$

Step 1.2

The first derivative of

f (x)

$f (x)$ with respect to

x

$x$ is

2 x + 3

$2 x + 3$ .

2 x + 3

$2 x + 3$

2 x + 3

$2 x + 3$

Step 2

Find if the derivative is continuous on

(- 3, 4)

$(- 3, 4)$ .

Tap for more steps...

Step 2.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

(- \infty, \infty)

$(- \infty, \infty)$

Set-Builder Notation:

{x | x \in ℝ}

${x | x \in ℝ}$

Step 2.2

f' (x)

$f' (x)$ is continuous on

(- 3, 4)

$(- 3, 4)$ .

The function is continuous.

Step 3

The function is differentiable on

(- 3, 4)

$(- 3, 4)$ because the derivative is continuous on

(- 3, 4)

$(- 3, 4)$ .

The function is differentiable.

Step 4

Enter YOUR Problem