Algebra Examples

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

Step 1

Differentiate both sides of the equation.

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

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

$6 x' + 7 i \cdot 1$

Step 2.3.3

Multiply $7$ by $1$ .

$6 x' + 7 i$

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 $y$ is $\frac{d}{d y} [18] + \frac{d}{d y} [- 21 i]$ .

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

Step 3.2

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

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

Step 3.3

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

$6 x' + 7 i = 0$

Step 5

Solve for $x'$ .

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Step 5.1

Subtract $7 i$ from both sides of the equation.

$6 x' = - 7 i$

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $6$ .

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Step 5.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.1.2

Divide $x'$ by $1$ .

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

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Move the negative in front of the fraction.

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

Step 6

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

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