Calculus Examples

$x y - 10 x + 8 = 0$

Step 1

Solve the equation as $y$ in terms of $x$ .

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Step 1.1

Move all terms not containing $y$ to the right side of the equation.

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Step 1.1.1

Add $10 x$ to both sides of the equation.

$x y + 8 = 10 x$

Step 1.1.2

Subtract $8$ from both sides of the equation.

$x y = 10 x - 8$

Step 1.2

Divide each term in $x y = 10 x - 8$ by $x$ and simplify.

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Step 1.2.1

Divide each term in $x y = 10 x - 8$ by $x$ .

$\frac{x y}{x} = \frac{10 x}{x} + \frac{- 8}{x}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $x$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{x y}{x} = \frac{10 x}{x} + \frac{- 8}{x}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{10 x}{x} + \frac{- 8}{x}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Cancel the common factor of $x$ .

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Step 1.2.3.1.1.1

Cancel the common factor.

$y = \frac{10 x}{x} + \frac{- 8}{x}$

Step 1.2.3.1.1.2

Divide $10$ by $1$ .

$y = 10 + \frac{- 8}{x}$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$y = 10 - \frac{8}{x}$

Step 2

Set $y$ as a function of $x$ .

$f (x) = 10 - \frac{8}{x}$

Step 3

Find the derivative.

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Step 3.1

Differentiate.

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Step 3.1.1

By the Sum Rule, the derivative of $10 - \frac{8}{x}$ with respect to $x$ is $\frac{d}{d x} [10] + \frac{d}{d x} [- \frac{8}{x}]$ .

$\frac{d}{d x} [10] + \frac{d}{d x} [- \frac{8}{x}]$

Step 3.1.2

Since $10$ is constant with respect to $x$ , the derivative of $10$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- \frac{8}{x}]$

Step 3.2

Evaluate $\frac{d}{d x} [- \frac{8}{x}]$ .

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Step 3.2.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- \frac{8}{x}$ with respect to $x$ is $- 8 \frac{d}{d x} [\frac{1}{x}]$ .

$0 - 8 \frac{d}{d x} [\frac{1}{x}]$

Step 3.2.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$0 - 8 \frac{d}{d x} [x^{- 1}]$

Step 3.2.3

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

$0 - 8 (- x^{- 2})$

Step 3.2.4

Multiply $- 1$ by $- 8$ .

$0 + 8 x^{- 2}$

Step 3.3

Simplify.

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Step 3.3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$0 + 8 \frac{1}{x^{2}}$

Step 3.3.2

Combine terms.

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Step 3.3.2.1

Combine $8$ and $\frac{1}{x^{2}}$ .

$0 + \frac{8}{x^{2}}$

Step 3.3.2.2

Add $0$ and $\frac{8}{x^{2}}$ .

$\frac{8}{x^{2}}$

Step 4

Set the derivative equal to $0$ then solve the equation $\frac{8}{x^{2}} = 0$ .

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Step 4.1

Set the numerator equal to zero.

$8 = 0$

Step 4.2

Since $8 \neq 0$ , there are no solutions.

No solution

Step 5

There are no solution found by setting the derivative equal to $0$ , $\frac{8}{x^{2}} = 0$ so there are no horizontal tangent lines.

No horizontal tangent lines found

Step 6