Calculus Examples

(3 i^{4} - 2 i^{2} + 5 i - 1) - (5 i^{3} + 4 i^{2} - i + 2)

$(3 i^{4} - 2 i^{2} + 5 i - 1) - (5 i^{3} + 4 i^{2} - i + 2)$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1

Rewrite

i^{4}

$i^{4}$ as

1

$1$ .

Tap for more steps...

Step 1.1.1.1

Rewrite

i^{4}

$i^{4}$ as

{(i^{2})}^{2}

${(i^{2})}^{2}$ .

\frac{d}{d x} [3 {(i^{2})}^{2} - 2 i^{2} + 5 i - 1 - (5 i^{3} + 4 i^{2} - i + 2)]

$\frac{d}{d x} [3 {(i^{2})}^{2} - 2 i^{2} + 5 i - 1 - (5 i^{3} + 4 i^{2} - i + 2)]$

Step 1.1.1.2

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

\frac{d}{d x} [3 {(- 1)}^{2} - 2 i^{2} + 5 i - 1 - (5 i^{3} + 4 i^{2} - i + 2)]

$\frac{d}{d x} [3 {(- 1)}^{2} - 2 i^{2} + 5 i - 1 - (5 i^{3} + 4 i^{2} - i + 2)]$

Step 1.1.1.3

Raise

- 1

$- 1$ to the power of

2

$2$ .

\frac{d}{d x} [3 \cdot 1 - 2 i^{2} + 5 i - 1 - (5 i^{3} + 4 i^{2} - i + 2)]

$\frac{d}{d x} [3 \cdot 1 - 2 i^{2} + 5 i - 1 - (5 i^{3} + 4 i^{2} - i + 2)]$

\frac{d}{d x} [3 \cdot 1 - 2 i^{2} + 5 i - 1 - (5 i^{3} + 4 i^{2} - i + 2)]

$\frac{d}{d x} [3 \cdot 1 - 2 i^{2} + 5 i - 1 - (5 i^{3} + 4 i^{2} - i + 2)]$

Step 1.1.2

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

\frac{d}{d x} [3 \cdot 1 - 2 \cdot - 1 + 5 i - 1 - (5 i^{3} + 4 i^{2} - i + 2)]

$\frac{d}{d x} [3 \cdot 1 - 2 \cdot - 1 + 5 i - 1 - (5 i^{3} + 4 i^{2} - i + 2)]$

Step 1.1.3

Differentiate using the Sum Rule.

Tap for more steps...

Step 1.1.3.1

Simplify each term.

Tap for more steps...

Step 1.1.3.1.1

Factor out

i^{2}

$i^{2}$ .

\frac{d}{d x} [3 \cdot 1 - 2 \cdot - 1 + 5 i - 1 - (5 (i^{2} \cdot i) + 4 i^{2} - i + 2)]

$\frac{d}{d x} [3 \cdot 1 - 2 \cdot - 1 + 5 i - 1 - (5 (i^{2} \cdot i) + 4 i^{2} - i + 2)]$

Step 1.1.3.1.2

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

\frac{d}{d x} [3 \cdot 1 - 2 \cdot - 1 + 5 i - 1 - (5 (- 1 \cdot i) + 4 i^{2} - i + 2)]

$\frac{d}{d x} [3 \cdot 1 - 2 \cdot - 1 + 5 i - 1 - (5 (- 1 \cdot i) + 4 i^{2} - i + 2)]$

Step 1.1.3.1.3

Rewrite

- 1 i

$- 1 i$ as

- i

$- i$ .

\frac{d}{d x} [3 \cdot 1 - 2 \cdot - 1 + 5 i - 1 - (5 (- i) + 4 i^{2} - i + 2)]

$\frac{d}{d x} [3 \cdot 1 - 2 \cdot - 1 + 5 i - 1 - (5 (- i) + 4 i^{2} - i + 2)]$

Step 1.1.3.1.4

Multiply

- 1

$- 1$ by

5

$5$ .

\frac{d}{d x} [3 \cdot 1 - 2 \cdot - 1 + 5 i - 1 - (- 5 i + 4 i^{2} - i + 2)]

$\frac{d}{d x} [3 \cdot 1 - 2 \cdot - 1 + 5 i - 1 - (- 5 i + 4 i^{2} - i + 2)]$

Step 1.1.3.1.5

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

\frac{d}{d x} [3 \cdot 1 - 2 \cdot - 1 + 5 i - 1 - (- 5 i + 4 \cdot - 1 - i + 2)]

$\frac{d}{d x} [3 \cdot 1 - 2 \cdot - 1 + 5 i - 1 - (- 5 i + 4 \cdot - 1 - i + 2)]$

Step 1.1.3.1.6

Multiply

4

$4$ by

- 1

$- 1$ .

\frac{d}{d x} [3 \cdot 1 - 2 \cdot - 1 + 5 i - 1 - (- 5 i - 4 - i + 2)]

$\frac{d}{d x} [3 \cdot 1 - 2 \cdot - 1 + 5 i - 1 - (- 5 i - 4 - i + 2)]$

\frac{d}{d x} [3 \cdot 1 - 2 \cdot - 1 + 5 i - 1 - (- 5 i - 4 - i + 2)]

$\frac{d}{d x} [3 \cdot 1 - 2 \cdot - 1 + 5 i - 1 - (- 5 i - 4 - i + 2)]$

Step 1.1.3.2

Simplify by adding terms.

Tap for more steps...

Step 1.1.3.2.1

Subtract

i

$i$ from

- 5 i

$- 5 i$ .

\frac{d}{d x} [3 \cdot 1 - 2 \cdot - 1 + 5 i - 1 - (- 4 - 6 i + 2)]

$\frac{d}{d x} [3 \cdot 1 - 2 \cdot - 1 + 5 i - 1 - (- 4 - 6 i + 2)]$

Step 1.1.3.2.2

Add

- 4

$- 4$ and

2

$2$ .

\frac{d}{d x} [3 \cdot 1 - 2 \cdot - 1 + 5 i - 1 - (- 2 - 6 i)]

$\frac{d}{d x} [3 \cdot 1 - 2 \cdot - 1 + 5 i - 1 - (- 2 - 6 i)]$

\frac{d}{d x} [3 \cdot 1 - 2 \cdot - 1 + 5 i - 1 - (- 2 - 6 i)]

$\frac{d}{d x} [3 \cdot 1 - 2 \cdot - 1 + 5 i - 1 - (- 2 - 6 i)]$

Step 1.1.3.3

By the Sum Rule, the derivative of

3 \cdot 1 - 2 \cdot - 1 + 5 i - 1 - (- 2 - 6 i)

$3 \cdot 1 - 2 \cdot - 1 + 5 i - 1 - (- 2 - 6 i)$ with respect to

x

$x$ is

\frac{d}{d x} [3 \cdot 1] + \frac{d}{d x} [- 2 \cdot - 1] + \frac{d}{d x} [5 i] + \frac{d}{d x} [- 1] + \frac{d}{d x} [- (- 2 - 6 i)]

$\frac{d}{d x} [3 \cdot 1] + \frac{d}{d x} [- 2 \cdot - 1] + \frac{d}{d x} [5 i] + \frac{d}{d x} [- 1] + \frac{d}{d x} [- (- 2 - 6 i)]$ .

\frac{d}{d x} [3 \cdot 1] + \frac{d}{d x} [- 2 \cdot - 1] + \frac{d}{d x} [5 i] + \frac{d}{d x} [- 1] + \frac{d}{d x} [- (- 2 - 6 i)]

$\frac{d}{d x} [3 \cdot 1] + \frac{d}{d x} [- 2 \cdot - 1] + \frac{d}{d x} [5 i] + \frac{d}{d x} [- 1] + \frac{d}{d x} [- (- 2 - 6 i)]$

\frac{d}{d x} [3 \cdot 1] + \frac{d}{d x} [- 2 \cdot - 1] + \frac{d}{d x} [5 i] + \frac{d}{d x} [- 1] + \frac{d}{d x} [- (- 2 - 6 i)]

$\frac{d}{d x} [3 \cdot 1] + \frac{d}{d x} [- 2 \cdot - 1] + \frac{d}{d x} [5 i] + \frac{d}{d x} [- 1] + \frac{d}{d x} [- (- 2 - 6 i)]$

Step 1.1.4

Evaluate

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

$\frac{d}{d x} [3 \cdot 1]$ .

Tap for more steps...

Step 1.1.4.1

Multiply

3

$3$ by

1

$1$ .

\frac{d}{d x} [3] + \frac{d}{d x} [- 2 \cdot - 1] + \frac{d}{d x} [5 i] + \frac{d}{d x} [- 1] + \frac{d}{d x} [- (- 2 - 6 i)]

$\frac{d}{d x} [3] + \frac{d}{d x} [- 2 \cdot - 1] + \frac{d}{d x} [5 i] + \frac{d}{d x} [- 1] + \frac{d}{d x} [- (- 2 - 6 i)]$

Step 1.1.4.2

Since

3

$3$ is constant with respect to

x

$x$ , the derivative of

3

$3$ with respect to

x

$x$ is

0

$0$ .

0 + \frac{d}{d x} [- 2 \cdot - 1] + \frac{d}{d x} [5 i] + \frac{d}{d x} [- 1] + \frac{d}{d x} [- (- 2 - 6 i)]

$0 + \frac{d}{d x} [- 2 \cdot - 1] + \frac{d}{d x} [5 i] + \frac{d}{d x} [- 1] + \frac{d}{d x} [- (- 2 - 6 i)]$

0 + \frac{d}{d x} [- 2 \cdot - 1] + \frac{d}{d x} [5 i] + \frac{d}{d x} [- 1] + \frac{d}{d x} [- (- 2 - 6 i)]

$0 + \frac{d}{d x} [- 2 \cdot - 1] + \frac{d}{d x} [5 i] + \frac{d}{d x} [- 1] + \frac{d}{d x} [- (- 2 - 6 i)]$

Step 1.1.5

Evaluate

\frac{d}{d x} [- 2 \cdot - 1]

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

Tap for more steps...

Step 1.1.5.1

Multiply

- 2

$- 2$ by

- 1

$- 1$ .

0 + \frac{d}{d x} [2] + \frac{d}{d x} [5 i] + \frac{d}{d x} [- 1] + \frac{d}{d x} [- (- 2 - 6 i)]

$0 + \frac{d}{d x} [2] + \frac{d}{d x} [5 i] + \frac{d}{d x} [- 1] + \frac{d}{d x} [- (- 2 - 6 i)]$

Step 1.1.5.2

Since

2

$2$ is constant with respect to

x

$x$ , the derivative of

2

$2$ with respect to

x

$x$ is

0

$0$ .

0 + 0 + \frac{d}{d x} [5 i] + \frac{d}{d x} [- 1] + \frac{d}{d x} [- (- 2 - 6 i)]

$0 + 0 + \frac{d}{d x} [5 i] + \frac{d}{d x} [- 1] + \frac{d}{d x} [- (- 2 - 6 i)]$

0 + 0 + \frac{d}{d x} [5 i] + \frac{d}{d x} [- 1] + \frac{d}{d x} [- (- 2 - 6 i)]

$0 + 0 + \frac{d}{d x} [5 i] + \frac{d}{d x} [- 1] + \frac{d}{d x} [- (- 2 - 6 i)]$

Step 1.1.6

Differentiate using the Constant Rule.

Tap for more steps...

Step 1.1.6.1

Since

5 i

$5 i$ is constant with respect to

x

$x$ , the derivative of

5 i

$5 i$ with respect to

x

$x$ is

0

$0$ .

0 + 0 + 0 + \frac{d}{d x} [- 1] + \frac{d}{d x} [- (- 2 - 6 i)]

$0 + 0 + 0 + \frac{d}{d x} [- 1] + \frac{d}{d x} [- (- 2 - 6 i)]$

Step 1.1.6.2

Since

- 1

$- 1$ is constant with respect to

x

$x$ , the derivative of

- 1

$- 1$ with respect to

x

$x$ is

0

$0$ .

0 + 0 + 0 + 0 + \frac{d}{d x} [- (- 2 - 6 i)]

$0 + 0 + 0 + 0 + \frac{d}{d x} [- (- 2 - 6 i)]$

Step 1.1.6.3

Since

- (- 2 - 6 i)

$- (- 2 - 6 i)$ is constant with respect to

x

$x$ , the derivative of

- (- 2 - 6 i)

$- (- 2 - 6 i)$ with respect to

x

$x$ is

0

$0$ .

0 + 0 + 0 + 0 + 0

$0 + 0 + 0 + 0 + 0$

0 + 0 + 0 + 0 + 0

$0 + 0 + 0 + 0 + 0$

Step 1.1.7

Combine terms.

Tap for more steps...

Step 1.1.7.1

Add

0

$0$ and

0

$0$ .

0 + 0 + 0 + 0

$0 + 0 + 0 + 0$

Step 1.1.7.2

Add

0

$0$ and

0

$0$ .

0 + 0 + 0

$0 + 0 + 0$

Step 1.1.7.3

Add

0

$0$ and

0

$0$ .

0 + 0

$0 + 0$

Step 1.1.7.4

Add

0

$0$ and

0

$0$ .

$f' (x) = 0$

Step 1.2

The first derivative of

$f (x)$ with respect to

$x$ is

$0$ .

$0$

Step 2

Set the first derivative equal to

$0$ then solve the equation

$0 = 0$ .

Tap for more steps...

Step 2.1

Set the first derivative equal to

$0$ .

$0 = 0$

Step 2.2

Since

$0 = 0$ , the equation will always be true.

Always true

Step 3

There are no values of

$x$ in the domain of the original problem where the derivative is

$0$ or undefined.

No critical points found