Calculus Examples

$x^{\frac{19}{9}} + x^{\frac{10}{9}}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

By the Sum Rule, the derivative of $x^{\frac{19}{9}} + x^{\frac{10}{9}}$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{19}{9}}] + \frac{d}{d x} [x^{\frac{10}{9}}]$ .

$\frac{d}{d x} [x^{\frac{19}{9}}] + \frac{d}{d x} [x^{\frac{10}{9}}]$

Step 1.1.2

Evaluate $\frac{d}{d x} [x^{\frac{19}{9}}]$ .

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Step 1.1.2.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{19}{9}$ .

$\frac{19}{9} x^{\frac{19}{9} - 1} + \frac{d}{d x} [x^{\frac{10}{9}}]$

Step 1.1.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{19}{9} x^{\frac{19}{9} - 1 \cdot \frac{9}{9}} + \frac{d}{d x} [x^{\frac{10}{9}}]$

Step 1.1.2.3

Combine $- 1$ and $\frac{9}{9}$ .

$\frac{19}{9} x^{\frac{19}{9} + \frac{- 1 \cdot 9}{9}} + \frac{d}{d x} [x^{\frac{10}{9}}]$

Step 1.1.2.4

Combine the numerators over the common denominator.

$\frac{19}{9} x^{\frac{19 - 1 \cdot 9}{9}} + \frac{d}{d x} [x^{\frac{10}{9}}]$

Step 1.1.2.5

Simplify the numerator.

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Step 1.1.2.5.1

Multiply $- 1$ by $9$ .

$\frac{19}{9} x^{\frac{19 - 9}{9}} + \frac{d}{d x} [x^{\frac{10}{9}}]$

Step 1.1.2.5.2

Subtract $9$ from $19$ .

$\frac{19}{9} x^{\frac{10}{9}} + \frac{d}{d x} [x^{\frac{10}{9}}]$

Step 1.1.3

Evaluate $\frac{d}{d x} [x^{\frac{10}{9}}]$ .

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Step 1.1.3.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{10}{9}$ .

$\frac{19}{9} x^{\frac{10}{9}} + \frac{10}{9} x^{\frac{10}{9} - 1}$

Step 1.1.3.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{19}{9} x^{\frac{10}{9}} + \frac{10}{9} x^{\frac{10}{9} - 1 \cdot \frac{9}{9}}$

Step 1.1.3.3

Combine $- 1$ and $\frac{9}{9}$ .

$\frac{19}{9} x^{\frac{10}{9}} + \frac{10}{9} x^{\frac{10}{9} + \frac{- 1 \cdot 9}{9}}$

Step 1.1.3.4

Combine the numerators over the common denominator.

$\frac{19}{9} x^{\frac{10}{9}} + \frac{10}{9} x^{\frac{10 - 1 \cdot 9}{9}}$

Step 1.1.3.5

Simplify the numerator.

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Step 1.1.3.5.1

Multiply $- 1$ by $9$ .

$\frac{19}{9} x^{\frac{10}{9}} + \frac{10}{9} x^{\frac{10 - 9}{9}}$

Step 1.1.3.5.2

Subtract $9$ from $10$ .

$\frac{19}{9} x^{\frac{10}{9}} + \frac{10}{9} x^{\frac{1}{9}}$

Step 1.1.4

Simplify each term.

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Step 1.1.4.1

Combine $\frac{19}{9}$ and $x^{\frac{10}{9}}$ .

$\frac{19 x^{\frac{10}{9}}}{9} + \frac{10}{9} x^{\frac{1}{9}}$

Step 1.1.4.2

Combine $\frac{10}{9}$ and $x^{\frac{1}{9}}$ .

$f' (x) = \frac{19 x^{\frac{10}{9}}}{9} + \frac{10 x^{\frac{1}{9}}}{9}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{19 x^{\frac{10}{9}}}{9} + \frac{10 x^{\frac{1}{9}}}{9}$ .

$\frac{19 x^{\frac{10}{9}}}{9} + \frac{10 x^{\frac{1}{9}}}{9}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{19 x^{\frac{10}{9}}}{9} + \frac{10 x^{\frac{1}{9}}}{9} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{19 x^{\frac{10}{9}}}{9} + \frac{10 x^{\frac{1}{9}}}{9} = 0$

Step 2.2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 0.52631578, 0$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

Evaluate $x^{\frac{19}{9}} + x^{\frac{10}{9}}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Substitute $- 0.52631578$ for $x$ .

${(- 0.52631578)}^{\frac{19}{9}} + {(- 0.52631578)}^{\frac{10}{9}}$

Step 4.2

Evaluate at $x = 0$ .

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Step 4.2.1

Substitute $0$ for $x$ .

${(0)}^{\frac{19}{9}} + {(0)}^{\frac{10}{9}}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Rewrite $0$ as $0^{9}$ .

${(0^{9})}^{\frac{19}{9}} + {(0)}^{\frac{10}{9}}$

Step 4.2.2.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$0^{9 (\frac{19}{9})} + {(0)}^{\frac{10}{9}}$

Step 4.2.2.1.3

Cancel the common factor of $9$ .

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Step 4.2.2.1.3.1

Cancel the common factor.

$0^{9 (\frac{19}{9})} + {(0)}^{\frac{10}{9}}$

Step 4.2.2.1.3.2

Rewrite the expression.

$0^{19} + {(0)}^{\frac{10}{9}}$

Step 4.2.2.1.4

Raising $0$ to any positive power yields $0$ .

$0 + {(0)}^{\frac{10}{9}}$

Step 4.2.2.1.5

Rewrite $0$ as $0^{9}$ .

$0 + {(0^{9})}^{\frac{10}{9}}$

Step 4.2.2.1.6

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$0 + 0^{9 (\frac{10}{9})}$

Step 4.2.2.1.7

Cancel the common factor of $9$ .

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Step 4.2.2.1.7.1

Cancel the common factor.

$0 + 0^{9 (\frac{10}{9})}$

Step 4.2.2.1.7.2

Rewrite the expression.

$0 + 0^{10}$

Step 4.2.2.1.8

Raising $0$ to any positive power yields $0$ .

$0 + 0$

Step 4.2.2.2

Add $0$ and $0$ .

$0$

Step 4.3

List all of the points.

$(- 0.52631578, {(- 0.52631578)}^{\frac{19}{9}} + {(- 0.52631578)}^{\frac{10}{9}}), (0, 0)$

Step 5