Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Rewrite $i^{4}$ as $1$ .

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Step 1.1.1.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

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

Step 1.1.1.2

Rewrite $i^{2}$ as $- 1$ .

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

Step 1.1.1.3

Raise $- 1$ to the power of $2$ .

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

Step 1.1.2

Rewrite $i^{2}$ as $- 1$ .

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

Step 1.1.3

Differentiate using the Sum Rule.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Factor out $i^{2}$ .

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

Step 1.1.3.1.2

Rewrite $i^{2}$ as $- 1$ .

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

Step 1.1.3.1.3

Rewrite $- 1 i$ as $- i$ .

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

Step 1.1.3.1.4

Multiply $- 1$ by $5$ .

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

Step 1.1.3.1.5

Rewrite $i^{2}$ as $- 1$ .

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

Step 1.1.3.1.6

Multiply $4$ by $- 1$ .

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

Step 1.1.3.2

Simplify by adding terms.

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Step 1.1.3.2.1

Subtract $i$ from $- 5 i$ .

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

Step 1.1.3.2.2

Add $- 4$ and $2$ .

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

Step 1.1.3.3

By the Sum Rule, the derivative of $3 \cdot 1 - 2 \cdot - 1 + 5 i - 1 - (- 2 - 6 i)$ with respect to $x$ is $\frac{d}{d x} [3 \cdot 1] + \frac{d}{d x} [- 2 \cdot - 1] + \frac{d}{d x} [5 i] + \frac{d}{d x} [- 1] + \frac{d}{d x} [- (- 2 - 6 i)]$ .

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

Step 1.1.4

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

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Step 1.1.4.1

Multiply $3$ by $1$ .

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

Step 1.1.4.2

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

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

Step 1.1.5

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

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Step 1.1.5.1

Multiply $- 2$ by $- 1$ .

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

Step 1.1.5.2

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

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

Step 1.1.6

Differentiate using the Constant Rule.

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Step 1.1.6.1

Since $5 i$ is constant with respect to $x$ , the derivative of $5 i$ with respect to $x$ is $0$ .

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

Step 1.1.6.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

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

Step 1.1.6.3

Since $- (- 2 - 6 i)$ is constant with respect to $x$ , the derivative of $- (- 2 - 6 i)$ with respect to $x$ is $0$ .

$0 + 0 + 0 + 0 + 0$

Step 1.1.7

Combine terms.

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Step 1.1.7.1

Add $0$ and $0$ .

$0 + 0 + 0 + 0$

Step 1.1.7.2

Add $0$ and $0$ .

$0 + 0 + 0$

Step 1.1.7.3

Add $0$ and $0$ .

$0 + 0$

Step 1.1.7.4

Add $0$ and $0$ .

$f' (x) = 0$

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 $0 = 0$ .

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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