Calculus Examples

${(x - 1)}^{3}$

Step 1

Find the first derivative.

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Find the first derivative.

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Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = x - 1$ .

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

$f' (x) = \frac{d}{d u} (u^{3}) \frac{d}{d x} (x - 1)$

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$f' (x) = 3 u^{2} \frac{d}{d x} (x - 1)$

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

$f' (x) = 3 {(x - 1)}^{2} \frac{d}{d x} (x - 1)$

Differentiate.

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

$f' (x) = 3 {(x - 1)}^{2} (\frac{d}{d x} (x) + \frac{d}{d x} (- 1))$

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

$f' (x) = 3 {(x - 1)}^{2} (1 + \frac{d}{d x} (- 1))$

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

$f' (x) = 3 {(x - 1)}^{2} (1 + 0)$

Simplify the expression.

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Add $1$ and $0$ .

$f' (x) = 3 {(x - 1)}^{2} \cdot 1$

Multiply $3$ by $1$ .

$f' (x) = 3 {(x - 1)}^{2}$

The first derivative of $f (x)$ with respect to $x$ is $3 {(x - 1)}^{2}$ .

$3 {(x - 1)}^{2}$

Step 2

Set the first derivative equal to $0$ then solve the equation $3 {(x - 1)}^{2} = 0$ .

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Set the first derivative equal to $0$ .

$3 {(x - 1)}^{2} = 0$

Divide each term in $3 {(x - 1)}^{2} = 0$ by $3$ and simplify.

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Divide each term in $3 {(x - 1)}^{2} = 0$ by $3$ .

$\frac{3 {(x - 1)}^{2}}{3} = \frac{0}{3}$

Simplify the left side.

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Cancel the common factor of $3$ .

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Cancel the common factor.

$\frac{3 {(x - 1)}^{2}}{3} = \frac{0}{3}$

Divide ${(x - 1)}^{2}$ by $1$ .

${(x - 1)}^{2} = \frac{0}{3}$

Simplify the right side.

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Divide $0$ by $3$ .

${(x - 1)}^{2} = 0$

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

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Step 3

Find the values where the derivative is undefined.

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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 ${(x - 1)}^{3}$ at each $x$ value where the derivative is $0$ or undefined.

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Evaluate at $x = 1$ .

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Substitute $1$ for $x$ .

${((1) - 1)}^{3}$

Simplify.

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Subtract $1$ from $1$ .

$0^{3}$

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

$0$

List all of the points.

$(1, 0)$

Step 5