Calculus Examples

$x = 45 + 3 x + 5 y + \frac{x y}{10}$

Step 1

Solve for $y$ .

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Step 1.1

Rewrite the equation as $45 + 3 x + 5 y + \frac{x y}{10} = x$ .

$45 + 3 x + 5 y + \frac{x y}{10} = x$

Step 1.2

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

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Step 1.2.1

Subtract $45$ from both sides of the equation.

$3 x + 5 y + \frac{x y}{10} = x - 45$

Step 1.2.2

Subtract $3 x$ from both sides of the equation.

$5 y + \frac{x y}{10} = x - 45 - 3 x$

Step 1.2.3

Subtract $3 x$ from $x$ .

$5 y + \frac{x y}{10} = - 2 x - 45$

Step 1.3

Factor $y$ out of $5 y + \frac{x y}{10}$ .

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Step 1.3.1

Factor $y$ out of $5 y$ .

$y \cdot 5 + \frac{x y}{10} = - 2 x - 45$

Step 1.3.2

Factor $y$ out of $\frac{x y}{10}$ .

$y \cdot 5 + y (\frac{x}{10}) = - 2 x - 45$

Step 1.3.3

Factor $y$ out of $y \cdot 5 + y \frac{x}{10}$ .

$y (5 + \frac{x}{10}) = - 2 x - 45$

Step 1.4

Divide each term in $y (5 + \frac{x}{10}) = - 2 x - 45$ by $5 + \frac{x}{10}$ and simplify.

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Step 1.4.1

Divide each term in $y (5 + \frac{x}{10}) = - 2 x - 45$ by $5 + \frac{x}{10}$ .

$\frac{y (5 + \frac{x}{10})}{5 + \frac{x}{10}} = \frac{- 2 x}{5 + \frac{x}{10}} + \frac{- 45}{5 + \frac{x}{10}}$

Step 1.4.2

Simplify the left side.

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Step 1.4.2.1

Cancel the common factor of $5 + \frac{x}{10}$ .

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Step 1.4.2.1.1

Cancel the common factor.

$\frac{y (5 + \frac{x}{10})}{5 + \frac{x}{10}} = \frac{- 2 x}{5 + \frac{x}{10}} + \frac{- 45}{5 + \frac{x}{10}}$

Step 1.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 2 x}{5 + \frac{x}{10}} + \frac{- 45}{5 + \frac{x}{10}}$

Step 1.4.3

Simplify the right side.

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Step 1.4.3.1

Combine the numerators over the common denominator.

$y = \frac{- 2 x - 45}{5 + \frac{x}{10}}$

Step 1.4.3.2

Multiply the numerator and denominator of the fraction by $10$ .

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Step 1.4.3.2.1

Multiply $\frac{- 2 x - 45}{5 + \frac{x}{10}}$ by $\frac{10}{10}$ .

$y = \frac{10}{10} \cdot \frac{- 2 x - 45}{5 + \frac{x}{10}}$

Step 1.4.3.2.2

Combine.

$y = \frac{10 (- 2 x - 45)}{10 (5 + \frac{x}{10})}$

Step 1.4.3.3

Apply the distributive property.

$y = \frac{10 (- 2 x) + 10 \cdot - 45}{10 \cdot 5 + 10 \frac{x}{10}}$

Step 1.4.3.4

Cancel the common factor of $10$ .

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Step 1.4.3.4.1

Cancel the common factor.

$y = \frac{10 (- 2 x) + 10 \cdot - 45}{10 \cdot 5 + 10 \frac{x}{10}}$

Step 1.4.3.4.2

Rewrite the expression.

$y = \frac{10 (- 2 x) + 10 \cdot - 45}{10 \cdot 5 + x}$

Step 1.4.3.5

Simplify the numerator.

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Step 1.4.3.5.1

Multiply $- 2$ by $10$ .

$y = \frac{- 20 x + 10 \cdot - 45}{10 \cdot 5 + x}$

Step 1.4.3.5.2

Multiply $10$ by $- 45$ .

$y = \frac{- 20 x - 450}{10 \cdot 5 + x}$

Step 1.4.3.5.3

Factor $10$ out of $- 20 x - 450$ .

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Step 1.4.3.5.3.1

Factor $10$ out of $- 20 x$ .

$y = \frac{10 (- 2 x) - 450}{10 \cdot 5 + x}$

Step 1.4.3.5.3.2

Factor $10$ out of $- 450$ .

$y = \frac{10 (- 2 x) + 10 (- 45)}{10 \cdot 5 + x}$

Step 1.4.3.5.3.3

Factor $10$ out of $10 (- 2 x) + 10 (- 45)$ .

$y = \frac{10 (- 2 x - 45)}{10 \cdot 5 + x}$

Step 1.4.3.6

Simplify with factoring out.

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Step 1.4.3.6.1

Multiply $10$ by $5$ .

$y = \frac{10 (- 2 x - 45)}{50 + x}$

Step 1.4.3.6.2

Factor $- 1$ out of $- 2 x$ .

$y = \frac{10 (- (2 x) - 45)}{50 + x}$

Step 1.4.3.6.3

Rewrite $- 45$ as $- 1 (45)$ .

$y = \frac{10 (- (2 x) - 1 (45))}{50 + x}$

Step 1.4.3.6.4

Factor $- 1$ out of $- (2 x) - 1 (45)$ .

$y = \frac{10 (- (2 x + 45))}{50 + x}$

Step 1.4.3.6.5

Simplify the expression.

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Step 1.4.3.6.5.1

Rewrite $- (2 x + 45)$ as $- 1 (2 x + 45)$ .

$y = \frac{10 (- 1 (2 x + 45))}{50 + x}$

Step 1.4.3.6.5.2

Move the negative in front of the fraction.

$y = - \frac{10 (2 x + 45)}{50 + x}$

Step 2

Find the first derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

Since $- 10$ is constant with respect to $x$ , the derivative of $- \frac{10 (2 x + 45)}{50 + x}$ with respect to $x$ is $- 10 \frac{d}{d x} [\frac{2 x + 45}{50 + x}]$ .

$- 10 \frac{d}{d x} [\frac{2 x + 45}{50 + x}]$

Step 2.1.2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 2 x + 45$ and $g (x) = 50 + x$ .

$- 10 \frac{(50 + x) \frac{d}{d x} [2 x + 45] - (2 x + 45) \frac{d}{d x} [50 + x]}{{(50 + x)}^{2}}$

Step 2.1.3

Differentiate.

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Step 2.1.3.1

By the Sum Rule, the derivative of $2 x + 45$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [45]$ .

$- 10 \frac{(50 + x) (\frac{d}{d x} [2 x] + \frac{d}{d x} [45]) - (2 x + 45) \frac{d}{d x} [50 + x]}{{(50 + x)}^{2}}$

Step 2.1.3.2

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

$- 10 \frac{(50 + x) (2 \frac{d}{d x} [x] + \frac{d}{d x} [45]) - (2 x + 45) \frac{d}{d x} [50 + x]}{{(50 + x)}^{2}}$

Step 2.1.3.3

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

$- 10 \frac{(50 + x) (2 \cdot 1 + \frac{d}{d x} [45]) - (2 x + 45) \frac{d}{d x} [50 + x]}{{(50 + x)}^{2}}$

Step 2.1.3.4

Multiply $2$ by $1$ .

$- 10 \frac{(50 + x) (2 + \frac{d}{d x} [45]) - (2 x + 45) \frac{d}{d x} [50 + x]}{{(50 + x)}^{2}}$

Step 2.1.3.5

Since $45$ is constant with respect to $x$ , the derivative of $45$ with respect to $x$ is $0$ .

$- 10 \frac{(50 + x) (2 + 0) - (2 x + 45) \frac{d}{d x} [50 + x]}{{(50 + x)}^{2}}$

Step 2.1.3.6

Simplify the expression.

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Step 2.1.3.6.1

Add $2$ and $0$ .

$- 10 \frac{(50 + x) \cdot 2 - (2 x + 45) \frac{d}{d x} [50 + x]}{{(50 + x)}^{2}}$

Step 2.1.3.6.2

Move $2$ to the left of $50 + x$ .

$- 10 \frac{2 \cdot (50 + x) - (2 x + 45) \frac{d}{d x} [50 + x]}{{(50 + x)}^{2}}$

Step 2.1.3.7

By the Sum Rule, the derivative of $50 + x$ with respect to $x$ is $\frac{d}{d x} [50] + \frac{d}{d x} [x]$ .

$- 10 \frac{2 (50 + x) - (2 x + 45) (\frac{d}{d x} [50] + \frac{d}{d x} [x])}{{(50 + x)}^{2}}$

Step 2.1.3.8

Since $50$ is constant with respect to $x$ , the derivative of $50$ with respect to $x$ is $0$ .

$- 10 \frac{2 (50 + x) - (2 x + 45) (0 + \frac{d}{d x} [x])}{{(50 + x)}^{2}}$

Step 2.1.3.9

Add $0$ and $\frac{d}{d x} [x]$ .

$- 10 \frac{2 (50 + x) - (2 x + 45) \frac{d}{d x} [x]}{{(50 + x)}^{2}}$

Step 2.1.3.10

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

$- 10 \frac{2 (50 + x) - (2 x + 45) \cdot 1}{{(50 + x)}^{2}}$

Step 2.1.3.11

Combine fractions.

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Step 2.1.3.11.1

Multiply $- 1$ by $1$ .

$- 10 \frac{2 (50 + x) - (2 x + 45)}{{(50 + x)}^{2}}$

Step 2.1.3.11.2

Combine $- 10$ and $\frac{2 (50 + x) - (2 x + 45)}{{(50 + x)}^{2}}$ .

$\frac{- 10 (2 (50 + x) - (2 x + 45))}{{(50 + x)}^{2}}$

Step 2.1.3.11.3

Move the negative in front of the fraction.

$- \frac{10 (2 (50 + x) - (2 x + 45))}{{(50 + x)}^{2}}$

Step 2.1.4

Simplify.

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Step 2.1.4.1

Apply the distributive property.

$- \frac{10 (2 \cdot 50 + 2 x - (2 x + 45))}{{(50 + x)}^{2}}$

Step 2.1.4.2

Apply the distributive property.

$- \frac{10 (2 \cdot 50 + 2 x - (2 x) - 1 \cdot 45)}{{(50 + x)}^{2}}$

Step 2.1.4.3

Apply the distributive property.

$- \frac{10 (2 \cdot 50) + 10 (2 x) + 10 (- (2 x)) + 10 (- 1 \cdot 45)}{{(50 + x)}^{2}}$

Step 2.1.4.4

Simplify the numerator.

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Step 2.1.4.4.1

Simplify each term.

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Step 2.1.4.4.1.1

Multiply $10 (2 \cdot 50)$ .

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Step 2.1.4.4.1.1.1

Multiply $2$ by $50$ .

$- \frac{10 \cdot 100 + 10 (2 x) + 10 (- (2 x)) + 10 (- 1 \cdot 45)}{{(50 + x)}^{2}}$

Step 2.1.4.4.1.1.2

Multiply $10$ by $100$ .

$- \frac{1000 + 10 (2 x) + 10 (- (2 x)) + 10 (- 1 \cdot 45)}{{(50 + x)}^{2}}$

Step 2.1.4.4.1.2

Multiply $2$ by $10$ .

$- \frac{1000 + 20 x + 10 (- (2 x)) + 10 (- 1 \cdot 45)}{{(50 + x)}^{2}}$

Step 2.1.4.4.1.3

Multiply $2$ by $- 1$ .

$- \frac{1000 + 20 x + 10 (- 2 x) + 10 (- 1 \cdot 45)}{{(50 + x)}^{2}}$

Step 2.1.4.4.1.4

Multiply $- 2$ by $10$ .

$- \frac{1000 + 20 x - 20 x + 10 (- 1 \cdot 45)}{{(50 + x)}^{2}}$

Step 2.1.4.4.1.5

Multiply $10 (- 1 \cdot 45)$ .

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Step 2.1.4.4.1.5.1

Multiply $- 1$ by $45$ .

$- \frac{1000 + 20 x - 20 x + 10 \cdot - 45}{{(50 + x)}^{2}}$

Step 2.1.4.4.1.5.2

Multiply $10$ by $- 45$ .

$- \frac{1000 + 20 x - 20 x - 450}{{(50 + x)}^{2}}$

Step 2.1.4.4.2

Combine the opposite terms in $1000 + 20 x - 20 x - 450$ .

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Step 2.1.4.4.2.1

Subtract $20 x$ from $20 x$ .

$- \frac{1000 + 0 - 450}{{(50 + x)}^{2}}$

Step 2.1.4.4.2.2

Add $1000$ and $0$ .

$- \frac{1000 - 450}{{(50 + x)}^{2}}$

Step 2.1.4.4.3

Subtract $450$ from $1000$ .

$f' (x) = - \frac{550}{{(50 + x)}^{2}}$

Step 2.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{550}{{(50 + x)}^{2}}$ .

$- \frac{550}{{(50 + x)}^{2}}$

Step 3

Set the first derivative equal to $0$ then solve the equation $- \frac{550}{{(50 + x)}^{2}} = 0$ .

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Step 3.1

Set the first derivative equal to $0$ .

$- \frac{550}{{(50 + x)}^{2}} = 0$

Step 3.2

Set the numerator equal to zero.

$550 = 0$

Step 3.3

Since $550 \neq 0$ , there are no solutions.

No solution

Step 4

Find the values where the derivative is undefined.

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Step 4.1

Set the denominator in $\frac{550}{{(50 + x)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(50 + x)}^{2} = 0$

Step 4.2

Solve for $x$ .

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Step 4.2.1

Set the $50 + x$ equal to $0$ .

$50 + x = 0$

Step 4.2.2

Subtract $50$ from both sides of the equation.

$x = - 50$

Step 5

Evaluate $- \frac{10 (2 x + 45)}{50 + x}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 5.1

Evaluate at $x = - 50$ .

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Step 5.1.1

Substitute $- 50$ for $x$ .

$- \frac{10 (2 (- 50) + 45)}{50 - 50}$

Step 5.1.2

Simplify.

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Step 5.1.2.1

Remove parentheses.

$- \frac{10 (2 (- 50) + 45)}{50 - 50}$

Step 5.1.2.2

Subtract $50$ from $50$ .

$- \frac{10 (2 (- 50) + 45)}{0}$

Step 5.1.2.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 6

There are no values of $x$ in the domain of the original problem where the derivative is $0$ or undefined.

No critical points found