Algebra Examples

$y = \frac{- 16 x^{2}}{0.434 v^{2}} + 1.15 x + 8$

Step 1

Find the first derivative of the function.

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Step 1.1

Move the negative in front of the fraction.

$\frac{d}{d x} [- \frac{16 x^{2}}{0.434 v^{2}} + 1.15 x + 8]$

Step 1.2

Factor $16$ out of $16 x^{2}$ .

$\frac{d}{d x} [- \frac{16 (x^{2})}{0.434 v^{2}} + 1.15 x + 8]$

Step 1.3

Factor $0.434$ out of $0.434 v^{2}$ .

$\frac{d}{d x} [- \frac{16 (x^{2})}{0.434 (v^{2})} + 1.15 x + 8]$

Step 1.4

Separate fractions.

$\frac{d}{d x} [- (\frac{16}{0.434} \cdot \frac{x^{2}}{v^{2}}) + 1.15 x + 8]$

Step 1.5

Differentiate using the Sum Rule.

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Step 1.5.1

Divide $16$ by $0.434$ .

$\frac{d}{d x} [- (36.86635944 \frac{x^{2}}{v^{2}}) + 1.15 x + 8]$

Step 1.5.2

Combine $36.86635944$ and $\frac{x^{2}}{v^{2}}$ .

$\frac{d}{d x} [- \frac{36.86635944 x^{2}}{v^{2}} + 1.15 x + 8]$

Step 1.5.3

By the Sum Rule, the derivative of $- \frac{36.86635944 x^{2}}{v^{2}} + 1.15 x + 8$ with respect to $x$ is $\frac{d}{d x} [- \frac{36.86635944 x^{2}}{v^{2}}] + \frac{d}{d x} [1.15 x] + \frac{d}{d x} [8]$ .

$\frac{d}{d x} [- \frac{36.86635944 x^{2}}{v^{2}}] + \frac{d}{d x} [1.15 x] + \frac{d}{d x} [8]$

Step 1.6

Evaluate $\frac{d}{d x} [- \frac{36.86635944 x^{2}}{v^{2}}]$ .

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Step 1.6.1

Since $- \frac{36.86635944}{v^{2}}$ is constant with respect to $x$ , the derivative of $- \frac{36.86635944 x^{2}}{v^{2}}$ with respect to $x$ is $- \frac{36.86635944}{v^{2}} \frac{d}{d x} [x^{2}]$ .

$- \frac{36.86635944}{v^{2}} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [1.15 x] + \frac{d}{d x} [8]$

Step 1.6.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$- \frac{36.86635944}{v^{2}} (2 x) + \frac{d}{d x} [1.15 x] + \frac{d}{d x} [8]$

Step 1.6.3

Multiply $2$ by $- 1$ .

$- 2 \frac{36.86635944}{v^{2}} x + \frac{d}{d x} [1.15 x] + \frac{d}{d x} [8]$

Step 1.6.4

Combine $- 2$ and $\frac{36.86635944}{v^{2}}$ .

$\frac{- 2 \cdot 36.86635944}{v^{2}} x + \frac{d}{d x} [1.15 x] + \frac{d}{d x} [8]$

Step 1.6.5

Multiply $- 2$ by $36.86635944$ .

$\frac{- 73.73271889}{v^{2}} x + \frac{d}{d x} [1.15 x] + \frac{d}{d x} [8]$

Step 1.6.6

Combine $\frac{- 73.73271889}{v^{2}}$ and $x$ .

$\frac{- 73.73271889 x}{v^{2}} + \frac{d}{d x} [1.15 x] + \frac{d}{d x} [8]$

Step 1.6.7

Move the negative in front of the fraction.

$- \frac{73.73271889 x}{v^{2}} + \frac{d}{d x} [1.15 x] + \frac{d}{d x} [8]$

Step 1.7

Evaluate $\frac{d}{d x} [1.15 x]$ .

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Step 1.7.1

Since $1.15$ is constant with respect to $x$ , the derivative of $1.15 x$ with respect to $x$ is $1.15 \frac{d}{d x} [x]$ .

$- \frac{73.73271889 x}{v^{2}} + 1.15 \frac{d}{d x} [x] + \frac{d}{d x} [8]$

Step 1.7.2

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

$- \frac{73.73271889 x}{v^{2}} + 1.15 \cdot 1 + \frac{d}{d x} [8]$

Step 1.7.3

Multiply $1.15$ by $1$ .

$- \frac{73.73271889 x}{v^{2}} + 1.15 + \frac{d}{d x} [8]$

Step 1.8

Differentiate using the Constant Rule.

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Step 1.8.1

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$- \frac{73.73271889 x}{v^{2}} + 1.15 + 0$

Step 1.8.2

Add $- \frac{73.73271889 x}{v^{2}} + 1.15$ and $0$ .

$- \frac{73.73271889 x}{v^{2}} + 1.15$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $- \frac{73.73271889 x}{v^{2}} + 1.15$ with respect to $x$ is $\frac{d}{d x} [- \frac{73.73271889 x}{v^{2}}] + \frac{d}{d x} [1.15]$ .

$f'' (x) = \frac{d}{d x} (- \frac{73.73271889 x}{v^{2}}) + \frac{d}{d x} (1.15)$

Step 2.2

Evaluate $\frac{d}{d x} [- \frac{73.73271889 x}{v^{2}}]$ .

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Step 2.2.1

Since $- \frac{73.73271889}{v^{2}}$ is constant with respect to $x$ , the derivative of $- \frac{73.73271889 x}{v^{2}}$ with respect to $x$ is $- \frac{73.73271889}{v^{2}} \frac{d}{d x} [x]$ .

$f'' (x) = - \frac{73.73271889}{v^{2}} \cdot \frac{d}{d x} x + \frac{d}{d x} (1.15)$

Step 2.2.2

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

$f'' (x) = - \frac{73.73271889}{v^{2}} \cdot 1 + \frac{d}{d x} (1.15)$

Step 2.2.3

Multiply $- 1$ by $1$ .

$f'' (x) = - \frac{73.73271889}{v^{2}} + \frac{d}{d x} (1.15)$

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

Since $1.15$ is constant with respect to $x$ , the derivative of $1.15$ with respect to $x$ is $0$ .

$f'' (x) = - \frac{73.73271889}{v^{2}} + 0$

Step 2.3.2

Add $- \frac{73.73271889}{v^{2}}$ and $0$ .

$f'' (x) = - \frac{73.73271889}{v^{2}}$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$- \frac{73.73271889 x}{v^{2}} + 1.15 = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

Move the negative in front of the fraction.

$\frac{d}{d x} [- \frac{16 x^{2}}{0.434 v^{2}} + 1.15 x + 8]$

Step 4.1.2

Factor $16$ out of $16 x^{2}$ .

$\frac{d}{d x} [- \frac{16 (x^{2})}{0.434 v^{2}} + 1.15 x + 8]$

Step 4.1.3

Factor $0.434$ out of $0.434 v^{2}$ .

$\frac{d}{d x} [- \frac{16 (x^{2})}{0.434 (v^{2})} + 1.15 x + 8]$

Step 4.1.4

Separate fractions.

$\frac{d}{d x} [- (\frac{16}{0.434} \cdot \frac{x^{2}}{v^{2}}) + 1.15 x + 8]$

Step 4.1.5

Differentiate using the Sum Rule.

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Step 4.1.5.1

Divide $16$ by $0.434$ .

$\frac{d}{d x} [- (36.86635944 \frac{x^{2}}{v^{2}}) + 1.15 x + 8]$

Step 4.1.5.2

Combine $36.86635944$ and $\frac{x^{2}}{v^{2}}$ .

$\frac{d}{d x} [- \frac{36.86635944 x^{2}}{v^{2}} + 1.15 x + 8]$

Step 4.1.5.3

$\frac{d}{d x} [- \frac{36.86635944 x^{2}}{v^{2}}] + \frac{d}{d x} [1.15 x] + \frac{d}{d x} [8]$

Step 4.1.6

Evaluate $\frac{d}{d x} [- \frac{36.86635944 x^{2}}{v^{2}}]$ .

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Step 4.1.6.1

$- \frac{36.86635944}{v^{2}} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [1.15 x] + \frac{d}{d x} [8]$

Step 4.1.6.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$- \frac{36.86635944}{v^{2}} (2 x) + \frac{d}{d x} [1.15 x] + \frac{d}{d x} [8]$

Step 4.1.6.3

Multiply $2$ by $- 1$ .

$- 2 \frac{36.86635944}{v^{2}} x + \frac{d}{d x} [1.15 x] + \frac{d}{d x} [8]$

Step 4.1.6.4

Combine $- 2$ and $\frac{36.86635944}{v^{2}}$ .

$\frac{- 2 \cdot 36.86635944}{v^{2}} x + \frac{d}{d x} [1.15 x] + \frac{d}{d x} [8]$

Step 4.1.6.5

Multiply $- 2$ by $36.86635944$ .

$\frac{- 73.73271889}{v^{2}} x + \frac{d}{d x} [1.15 x] + \frac{d}{d x} [8]$

Step 4.1.6.6

Combine $\frac{- 73.73271889}{v^{2}}$ and $x$ .

$\frac{- 73.73271889 x}{v^{2}} + \frac{d}{d x} [1.15 x] + \frac{d}{d x} [8]$

Step 4.1.6.7

Move the negative in front of the fraction.

$- \frac{73.73271889 x}{v^{2}} + \frac{d}{d x} [1.15 x] + \frac{d}{d x} [8]$

Step 4.1.7

Evaluate $\frac{d}{d x} [1.15 x]$ .

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Step 4.1.7.1

Since $1.15$ is constant with respect to $x$ , the derivative of $1.15 x$ with respect to $x$ is $1.15 \frac{d}{d x} [x]$ .

$- \frac{73.73271889 x}{v^{2}} + 1.15 \frac{d}{d x} [x] + \frac{d}{d x} [8]$

Step 4.1.7.2

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

$- \frac{73.73271889 x}{v^{2}} + 1.15 \cdot 1 + \frac{d}{d x} [8]$

Step 4.1.7.3

Multiply $1.15$ by $1$ .

$- \frac{73.73271889 x}{v^{2}} + 1.15 + \frac{d}{d x} [8]$

Step 4.1.8

Differentiate using the Constant Rule.

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Step 4.1.8.1

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$- \frac{73.73271889 x}{v^{2}} + 1.15 + 0$

Step 4.1.8.2

Add $- \frac{73.73271889 x}{v^{2}} + 1.15$ and $0$ .

$f' (x) = - \frac{73.73271889 x}{v^{2}} + 1.15$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{73.73271889 x}{v^{2}} + 1.15$ .

$- \frac{73.73271889 x}{v^{2}} + 1.15$

Step 5

Set the first derivative equal to $0$ then solve the equation $- \frac{73.73271889 x}{v^{2}} + 1.15 = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$- \frac{73.73271889 x}{v^{2}} + 1.15 = 0$

Step 5.2

Subtract $1.15$ from both sides of the equation.

$- \frac{73.73271889 x}{v^{2}} = - 1.15$

Step 5.3

Divide each term in $- \frac{73.73271889 x}{v^{2}} = - 1.15$ by $- 1$ and simplify.

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Step 5.3.1

Divide each term in $- \frac{73.73271889 x}{v^{2}} = - 1.15$ by $- 1$ .

$\frac{- \frac{73.73271889 x}{v^{2}}}{- 1} = \frac{- 1.15}{- 1}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{73.73271889 x}{v^{2}}}{1} = \frac{- 1.15}{- 1}$

Step 5.3.2.2

Divide $\frac{73.73271889 x}{v^{2}}$ by $1$ .

$\frac{73.73271889 x}{v^{2}} = \frac{- 1.15}{- 1}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Divide $- 1.15$ by $- 1$ .

$\frac{73.73271889 x}{v^{2}} = 1.15$

Step 5.4

Multiply both sides by $v^{2}$ .

$\frac{73.73271889 x}{v^{2}} v^{2} = 1.15 v^{2}$

Step 5.5

Simplify the left side.

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Step 5.5.1

Cancel the common factor of $v^{2}$ .

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Step 5.5.1.1

Cancel the common factor.

$\frac{73.73271889 x}{v^{2}} v^{2} = 1.15 v^{2}$

Step 5.5.1.2

Rewrite the expression.

$73.73271889 x = 1.15 v^{2}$

Step 5.6

Divide each term in $73.73271889 x = 1.15 v^{2}$ by $73.73271889$ and simplify.

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Step 5.6.1

Divide each term in $73.73271889 x = 1.15 v^{2}$ by $73.73271889$ .

$\frac{73.73271889 x}{73.73271889} = \frac{1.15 v^{2}}{73.73271889}$

Step 5.6.2

Simplify the left side.

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Step 5.6.2.1

Cancel the common factor of $73.73271889$ .

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Step 5.6.2.1.1

Cancel the common factor.

$\frac{73.73271889 x}{73.73271889} = \frac{1.15 v^{2}}{73.73271889}$

Step 5.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{1.15 v^{2}}{73.73271889}$

Step 5.6.3

Simplify the right side.

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Step 5.6.3.1

Factor $1.15$ out of $1.15 v^{2}$ .

$x = \frac{1.15 (v^{2})}{73.73271889}$

Step 5.6.3.2

Factor $73.73271889$ out of $73.73271889$ .

$x = \frac{1.15 (v^{2})}{73.73271889 (1)}$

Step 5.6.3.3

Separate fractions.

$x = \frac{1.15}{73.73271889} \cdot \frac{v^{2}}{1}$

Step 5.6.3.4

Divide $1.15$ by $73.73271889$ .

$x = 0.01559687 \frac{v^{2}}{1}$

Step 5.6.3.5

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

$x = 0.01559687 v^{2}$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Set the denominator in $\frac{73.73271889 x}{v^{2}}$ equal to $0$ to find where the expression is undefined.

$v^{2} = 0$

Step 6.2

Solve for $x$ .

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Step 6.2.1

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

$v = \pm \sqrt{0}$

Step 6.2.2

Simplify $\pm \sqrt{0}$ .

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Step 6.2.2.1

Rewrite $0$ as $0^{2}$ .

$v = \pm \sqrt{0^{2}}$

Step 6.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$v = \pm 0$

Step 6.2.2.3

Plus or minus $0$ is $0$ .

$v = 0$

Step 6.3

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = 0$

Step 7

Critical points to evaluate.

$x = 0.01559687 v^{2}$

$x = 0$

Step 8

Evaluate the second derivative at $x = 0.01559687 v^{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{73.73271889}{v^{2}}$

Step 9

Since the first derivative test failed, there are no local extrema.

No Local Extrema

Step 10