Algebra Examples

$6 x + 7 i y = 18 - 21 i$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (6 x + 7 i y) = \frac{d}{d x} (18 - 21 i)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $6 x + 7 i y$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [7 i y]$ .

$\frac{d}{d x} [6 x] + \frac{d}{d x} [7 i y]$

Step 2.2

Evaluate $\frac{d}{d x} [6 x]$ .

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Step 2.2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$6 \frac{d}{d x} [x] + \frac{d}{d x} [7 i y]$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$6 \cdot 1 + \frac{d}{d x} [7 i y]$

Step 2.2.3

Multiply $6$ by $1$ .

$6 + \frac{d}{d x} [7 i y]$

Step 2.3

Evaluate $\frac{d}{d x} [7 i y]$ .

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Step 2.3.1

Since $7 i$ is constant with respect to $x$ , the derivative of $7 i y$ with respect to $x$ is $7 i \frac{d}{d x} [y]$ .

$6 + 7 i \frac{d}{d x} [y]$

Step 2.3.2

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$6 + 7 i y'$

Step 2.4

Reorder terms.

$7 i y' + 6$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $18 - 21 i$ with respect to $x$ is $\frac{d}{d x} [18] + \frac{d}{d x} [- 21 i]$ .

$\frac{d}{d x} [18] + \frac{d}{d x} [- 21 i]$

Step 3.2

Since $18$ is constant with respect to $x$ , the derivative of $18$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 21 i]$

Step 3.3

Since $- 21 i$ is constant with respect to $x$ , the derivative of $- 21 i$ with respect to $x$ is $0$ .

$0 + 0$

Step 3.4

Add $0$ and $0$ .

$0$

Step 4

Reform the equation by setting the left side equal to the right side.

$7 i y' + 6 = 0$

Step 5

Solve for $y'$ .

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Step 5.1

Subtract $6$ from both sides of the equation.

$7 i y' = - 6$

Step 5.2

Divide each term in $7 i y' = - 6$ by $7 i$ and simplify.

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Step 5.2.1

Divide each term in $7 i y' = - 6$ by $7 i$ .

$\frac{7 i y'}{7 i} = \frac{- 6}{7 i}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $7$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{7 i y'}{7 i} = \frac{- 6}{7 i}$

Step 5.2.2.1.2

Rewrite the expression.

$\frac{i y'}{i} = \frac{- 6}{7 i}$

Step 5.2.2.2

Cancel the common factor of $i$ .

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Step 5.2.2.2.1

Cancel the common factor.

$\frac{i y'}{i} = \frac{- 6}{7 i}$

Step 5.2.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{- 6}{7 i}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Multiply the numerator and denominator of $\frac{- 6}{7 i}$ by the conjugate of $7 i$ to make the denominator real.

$y' = \frac{- 6}{7 i} \cdot \frac{i}{i}$

Step 5.2.3.2

Multiply.

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Step 5.2.3.2.1

Combine.

$y' = \frac{- 6 i}{7 i i}$

Step 5.2.3.2.2

Simplify the denominator.

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Step 5.2.3.2.2.1

Add parentheses.

$y' = \frac{- 6 i}{7 (i i)}$

Step 5.2.3.2.2.2

Raise $i$ to the power of $1$ .

$y' = \frac{- 6 i}{7 (i^{1} i)}$

Step 5.2.3.2.2.3

Raise $i$ to the power of $1$ .

$y' = \frac{- 6 i}{7 (i^{1} i^{1})}$

Step 5.2.3.2.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y' = \frac{- 6 i}{7 i^{1 + 1}}$

Step 5.2.3.2.2.5

Add $1$ and $1$ .

$y' = \frac{- 6 i}{7 i^{2}}$

Step 5.2.3.2.2.6

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

$y' = \frac{- 6 i}{7 \cdot - 1}$

Step 5.2.3.3

Multiply $7$ by $- 1$ .

$y' = \frac{- 6 i}{- 7}$

Step 5.2.3.4

Dividing two negative values results in a positive value.

$y' = \frac{6 i}{7}$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{6 i}{7}$