Algebra Examples

$z_{x} = 2 y^{2} x + 2 x + 2 y$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (z_{x}) = \frac{d}{d x} (2 y^{2} x + 2 x + 2 y)$

Step 2

Since $z_{x}$ is constant with respect to $x$ , the derivative of $z_{x}$ with respect to $x$ is $0$ .

$0$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $2 y^{2} x + 2 x + 2 y$ with respect to $x$ is $\frac{d}{d x} [2 y^{2} x] + \frac{d}{d x} [2 x] + \frac{d}{d x} [2 y]$ .

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

Step 3.2

Evaluate $\frac{d}{d x} [2 y^{2} x]$ .

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Step 3.2.1

Since $2 y^{2}$ is constant with respect to $x$ , the derivative of $2 y^{2} x$ with respect to $x$ is $2 y^{2} \frac{d}{d x} [x]$ .

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

Step 3.2.2

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

$2 y^{2} \cdot 1 + \frac{d}{d x} [2 x] + \frac{d}{d x} [2 y]$

Step 3.2.3

Multiply $2$ by $1$ .

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

Step 3.3

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

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Step 3.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 3.3.2

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

$2 y^{2} + 2 \cdot 1 + \frac{d}{d x} [2 y]$

Step 3.3.3

Multiply $2$ by $1$ .

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

Step 3.4

Differentiate using the Constant Rule.

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Step 3.4.1

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

$2 y^{2} + 2 + 0$

Step 3.4.2

Add $2 y^{2} + 2$ and $0$ .

$2 y^{2} + 2$

Step 4

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

$0 = 2 y^{2} + 2$

Step 5

Solve for $y$ .

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Step 5.1

Since $y$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$2 y^{2} + 2 = 0$

Step 5.2

Subtract $2$ from both sides of the equation.

$2 y^{2} = - 2$

Step 5.3

Divide each term in $2 y^{2} = - 2$ by $2$ and simplify.

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Step 5.3.1

Divide each term in $2 y^{2} = - 2$ by $2$ .

$\frac{2 y^{2}}{2} = \frac{- 2}{2}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $2$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{2 y^{2}}{2} = \frac{- 2}{2}$

Step 5.3.2.1.2

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{- 2}{2}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Divide $- 2$ by $2$ .

$y^{2} = - 1$

Step 5.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{- 1}$

Step 5.5

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \pm i$

Step 5.6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.6.1

First, use the positive value of the $\pm$ to find the first solution.

$y = i$

Step 5.6.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - i$

Step 5.6.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = i, - i$

Step 6

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

$\frac{d z}{d x} = i, - i$