Calculus Examples

$f (x) = x^{8} {(x - 1)}^{7}$

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

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{8}$ and $g (x) = {(x - 1)}^{7}$ .

$f' (x) = x^{8} \frac{d}{d x} ({(x - 1)}^{7}) + {(x - 1)}^{7} \frac{d}{d x} (x^{8})$

Step 1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{7}$ and $g (x) = x - 1$ .

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Step 1.1.2.1

To apply the Chain Rule, set $u$ as $x - 1$ .

$f' (x) = x^{8} (\frac{d}{d u} (u^{7}) \frac{d}{d x} (x - 1)) + {(x - 1)}^{7} \frac{d}{d x} (x^{8})$

Step 1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 7$ .

$f' (x) = x^{8} (7 u^{6} \frac{d}{d x} (x - 1)) + {(x - 1)}^{7} \frac{d}{d x} (x^{8})$

Step 1.1.2.3

Replace all occurrences of $u$ with $x - 1$ .

$f' (x) = x^{8} (7 {(x - 1)}^{6} \frac{d}{d x} (x - 1)) + {(x - 1)}^{7} \frac{d}{d x} (x^{8})$

Step 1.1.3

Differentiate.

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Step 1.1.3.1

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

$f' (x) = x^{8} (7 {(x - 1)}^{6} (\frac{d}{d x} (x) + \frac{d}{d x} (- 1))) + {(x - 1)}^{7} \frac{d}{d x} (x^{8})$

Step 1.1.3.2

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

$f' (x) = x^{8} (7 {(x - 1)}^{6} (1 + \frac{d}{d x} (- 1))) + {(x - 1)}^{7} \frac{d}{d x} (x^{8})$

Step 1.1.3.3

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$f' (x) = x^{8} (7 {(x - 1)}^{6} (1 + 0)) + {(x - 1)}^{7} \frac{d}{d x} (x^{8})$

Step 1.1.3.4

Simplify the expression.

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Step 1.1.3.4.1

Add $1$ and $0$ .

$f' (x) = x^{8} (7 {(x - 1)}^{6} \cdot 1) + {(x - 1)}^{7} \frac{d}{d x} (x^{8})$

Step 1.1.3.4.2

Multiply $7$ by $1$ .

$f' (x) = x^{8} (7 {(x - 1)}^{6}) + {(x - 1)}^{7} \frac{d}{d x} (x^{8})$

Step 1.1.3.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 8$ .

$f' (x) = x^{8} (7 {(x - 1)}^{6}) + {(x - 1)}^{7} (8 x^{7})$

Step 1.1.3.6

Move $8$ to the left of ${(x - 1)}^{7}$ .

$f' (x) = x^{8} (7 {(x - 1)}^{6}) + 8 \cdot ({(x - 1)}^{7} x^{7})$

Step 1.1.4

Simplify.

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Step 1.1.4.1

Factor $x^{7} {(x - 1)}^{6}$ out of $x^{8} (7 {(x - 1)}^{6}) + 8 {(x - 1)}^{7} x^{7}$ .

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Step 1.1.4.1.1

Factor $x^{7} {(x - 1)}^{6}$ out of $x^{8} (7 {(x - 1)}^{6})$ .

$f' (x) = x^{7} {(x - 1)}^{6} (x \cdot (7)) + 8 {(x - 1)}^{7} x^{7}$

Step 1.1.4.1.2

Factor $x^{7} {(x - 1)}^{6}$ out of $8 {(x - 1)}^{7} x^{7}$ .

$f' (x) = x^{7} {(x - 1)}^{6} (x \cdot (7)) + x^{7} {(x - 1)}^{6} (8 (x - 1))$

Step 1.1.4.1.3

Factor $x^{7} {(x - 1)}^{6}$ out of $x^{7} {(x - 1)}^{6} (x \cdot (7)) + x^{7} {(x - 1)}^{6} (8 (x - 1))$ .

$f' (x) = x^{7} {(x - 1)}^{6} (x \cdot (7) + 8 (x - 1))$

Step 1.1.4.2

Move $7$ to the left of $x$ .

$f' (x) = x^{7} {(x - 1)}^{6} (7 x + 8 (x - 1))$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $x^{7} {(x - 1)}^{6} (7 x + 8 (x - 1))$ .

$x^{7} {(x - 1)}^{6} (7 x + 8 (x - 1))$

Step 2

Set the first derivative equal to $0$ then solve the equation $x^{7} {(x - 1)}^{6} (7 x + 8 (x - 1)) = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$x^{7} {(x - 1)}^{6} (7 x + 8 (x - 1)) = 0$

Step 2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{7} = 0$

${(x - 1)}^{6} = 0$

$7 x + 8 (x - 1) = 0$

Step 2.3

Set $x^{7}$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $x^{7}$ equal to $0$ .

$x^{7} = 0$

Step 2.3.2

Solve $x^{7} = 0$ for $x$ .

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Step 2.3.2.1

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

$x = \sqrt[7]{0}$

Step 2.3.2.2

Simplify $\sqrt[7]{0}$ .

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Step 2.3.2.2.1

Rewrite $0$ as $0^{7}$ .

$x = \sqrt[7]{0^{7}}$

Step 2.3.2.2.2

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

$x = 0$

Step 2.4

Set ${(x - 1)}^{6}$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set ${(x - 1)}^{6}$ equal to $0$ .

${(x - 1)}^{6} = 0$

Step 2.4.2

Solve ${(x - 1)}^{6} = 0$ for $x$ .

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Step 2.4.2.1

Set the $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 2.4.2.2

Add $1$ to both sides of the equation.

$x = 1$

Step 2.5

Set $7 x + 8 (x - 1)$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $7 x + 8 (x - 1)$ equal to $0$ .

$7 x + 8 (x - 1) = 0$

Step 2.5.2

Solve $7 x + 8 (x - 1) = 0$ for $x$ .

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Step 2.5.2.1

Simplify $7 x + 8 (x - 1)$ .

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Step 2.5.2.1.1

Simplify each term.

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Step 2.5.2.1.1.1

Apply the distributive property.

$7 x + 8 x + 8 \cdot - 1 = 0$

Step 2.5.2.1.1.2

Multiply $8$ by $- 1$ .

$7 x + 8 x - 8 = 0$

Step 2.5.2.1.2

Add $7 x$ and $8 x$ .

$15 x - 8 = 0$

Step 2.5.2.2

Add $8$ to both sides of the equation.

$15 x = 8$

Step 2.5.2.3

Divide each term in $15 x = 8$ by $15$ and simplify.

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Step 2.5.2.3.1

Divide each term in $15 x = 8$ by $15$ .

$\frac{15 x}{15} = \frac{8}{15}$

Step 2.5.2.3.2

Simplify the left side.

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Step 2.5.2.3.2.1

Cancel the common factor of $15$ .

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Step 2.5.2.3.2.1.1

Cancel the common factor.

$\frac{15 x}{15} = \frac{8}{15}$

Step 2.5.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{8}{15}$

Step 2.6

The final solution is all the values that make $x^{7} {(x - 1)}^{6} (7 x + 8 (x - 1)) = 0$ true.

$x = 0, 1, \frac{8}{15}$

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 $x^{8} {(x - 1)}^{7}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 0$ .

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Step 4.1.1

Substitute $0$ for $x$ .

${(0)}^{8} {((0) - 1)}^{7}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

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

$0 {((0) - 1)}^{7}$

Step 4.1.2.2

Subtract $1$ from $0$ .

$0 {(- 1)}^{7}$

Step 4.1.2.3

Raise $- 1$ to the power of $7$ .

$0 \cdot - 1$

Step 4.1.2.4

Multiply $0$ by $- 1$ .

$0$

Step 4.2

Evaluate at $x = 1$ .

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Step 4.2.1

Substitute $1$ for $x$ .

${(1)}^{8} {((1) - 1)}^{7}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

One to any power is one.

$1 {((1) - 1)}^{7}$

Step 4.2.2.2

Multiply ${((1) - 1)}^{7}$ by $1$ .

${((1) - 1)}^{7}$

Step 4.2.2.3

Subtract $1$ from $1$ .

$0^{7}$

Step 4.2.2.4

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

$0$

Step 4.3

Evaluate at $x = \frac{8}{15}$ .

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Step 4.3.1

Substitute $\frac{8}{15}$ for $x$ .

${(\frac{8}{15})}^{8} {((\frac{8}{15}) - 1)}^{7}$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Apply the product rule to $\frac{8}{15}$ .

$\frac{8^{8}}{15^{8}} {((\frac{8}{15}) - 1)}^{7}$

Step 4.3.2.2

Raise $8$ to the power of $8$ .

$\frac{16777216}{15^{8}} {((\frac{8}{15}) - 1)}^{7}$

Step 4.3.2.3

Raise $15$ to the power of $8$ .

$\frac{16777216}{2562890625} {((\frac{8}{15}) - 1)}^{7}$

Step 4.3.2.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$\frac{16777216}{2562890625} {(\frac{8}{15} - 1 \cdot \frac{15}{15})}^{7}$

Step 4.3.2.5

Combine $- 1$ and $\frac{15}{15}$ .

$\frac{16777216}{2562890625} {(\frac{8}{15} + \frac{- 1 \cdot 15}{15})}^{7}$

Step 4.3.2.6

Combine the numerators over the common denominator.

$\frac{16777216}{2562890625} {(\frac{8 - 1 \cdot 15}{15})}^{7}$

Step 4.3.2.7

Simplify the numerator.

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Step 4.3.2.7.1

Multiply $- 1$ by $15$ .

$\frac{16777216}{2562890625} {(\frac{8 - 15}{15})}^{7}$

Step 4.3.2.7.2

Subtract $15$ from $8$ .

$\frac{16777216}{2562890625} {(\frac{- 7}{15})}^{7}$

Step 4.3.2.8

Move the negative in front of the fraction.

$\frac{16777216}{2562890625} {(- \frac{7}{15})}^{7}$

Step 4.3.2.9

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.3.2.9.1

Apply the product rule to $- \frac{7}{15}$ .

$\frac{16777216}{2562890625} ({(- 1)}^{7} {(\frac{7}{15})}^{7})$

Step 4.3.2.9.2

Apply the product rule to $\frac{7}{15}$ .

$\frac{16777216}{2562890625} ({(- 1)}^{7} \frac{7^{7}}{15^{7}})$

Step 4.3.2.10

Raise $- 1$ to the power of $7$ .

$\frac{16777216}{2562890625} (- \frac{7^{7}}{15^{7}})$

Step 4.3.2.11

Raise $7$ to the power of $7$ .

$\frac{16777216}{2562890625} (- \frac{823543}{15^{7}})$

Step 4.3.2.12

Raise $15$ to the power of $7$ .

$\frac{16777216}{2562890625} (- \frac{823543}{170859375})$

Step 4.3.2.13

Multiply $\frac{16777216}{2562890625} (- \frac{823543}{170859375})$ .

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Step 4.3.2.13.1

Multiply $\frac{16777216}{2562890625}$ by $\frac{823543}{170859375}$ .

$- \frac{16777216 \cdot 823543}{2562890625 \cdot 170859375}$

Step 4.3.2.13.2

Multiply $16777216$ by $823543$ .

$- \frac{13816758796288}{2562890625 \cdot 170859375}$

Step 4.3.2.13.3

Multiply $2562890625$ by $170859375$ .

$- \frac{13816758796288}{437893890380859375}$

Step 4.4

List all of the points.

$(0, 0), (1, 0), (\frac{8}{15}, - \frac{13816758796288}{437893890380859375})$

Step 5