Calculus Examples

$| 3 t - 4 |$

Find the first derivative.

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Find the first derivative.

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Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = | t |$ and $g (t) = 3 t - 4$ .

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To apply the Chain Rule, set $u$ as $3 t - 4$ .

$f' (t) = \frac{d}{d u} (| u |) \frac{d}{d t} (3 t - 4)$

The derivative of $| u |$ with respect to $u$ is $\frac{u}{| u |}$ .

$f' (t) = \frac{u}{| u |} \cdot \frac{d}{d t} (3 t - 4)$

Replace all occurrences of $u$ with $3 t - 4$ .

$f' (t) = \frac{3 t - 4}{| 3 t - 4 |} \cdot \frac{d}{d t} (3 t - 4)$

Differentiate.

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By the Sum Rule, the derivative of $3 t - 4$ with respect to $t$ is $\frac{d}{d t} [3 t] + \frac{d}{d t} [- 4]$ .

$f' (t) = \frac{3 t - 4}{| 3 t - 4 |} \cdot (\frac{d}{d t} (3 t) + \frac{d}{d t} (- 4))$

Since $3$ is constant with respect to $t$ , the derivative of $3 t$ with respect to $t$ is $3 \frac{d}{d t} [t]$ .

$f' (t) = \frac{3 t - 4}{| 3 t - 4 |} \cdot (3 \frac{d}{d t} (t) + \frac{d}{d t} (- 4))$

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

$f' (t) = \frac{3 t - 4}{| 3 t - 4 |} \cdot (3 \cdot 1 + \frac{d}{d t} (- 4))$

Multiply $3$ by $1$ .

$f' (t) = \frac{3 t - 4}{| 3 t - 4 |} \cdot (3 + \frac{d}{d t} (- 4))$

Since $- 4$ is constant with respect to $t$ , the derivative of $- 4$ with respect to $t$ is $0$ .

$f' (t) = \frac{3 t - 4}{| 3 t - 4 |} \cdot (3 + 0)$

Combine fractions.

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Add $3$ and $0$ .

$f' (t) = \frac{3 t - 4}{| 3 t - 4 |} \cdot 3$

Combine $\frac{3 t - 4}{| 3 t - 4 |}$ and $3$ .

$f' (t) = \frac{(3 t - 4) \cdot 3}{| 3 t - 4 |}$

Move $3$ to the left of $3 t - 4$ .

$f' (t) = \frac{3 \cdot (3 t - 4)}{| 3 t - 4 |}$

Simplify.

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Apply the distributive property.

$f' (t) = \frac{3 (3 t) + 3 \cdot - 4}{| 3 t - 4 |}$

Simplify each term.

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Multiply $3$ by $3$ .

$f' (t) = \frac{9 t + 3 \cdot - 4}{| 3 t - 4 |}$

Multiply $3$ by $- 4$ .

$f' (t) = \frac{9 t - 12}{| 3 t - 4 |}$

Factor $3$ out of $9 t - 12$ .

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Factor $3$ out of $9 t$ .

$f' (t) = \frac{3 (3 t) - 12}{| 3 t - 4 |}$

Factor $3$ out of $- 12$ .

$f' (t) = \frac{3 (3 t) + 3 (- 4)}{| 3 t - 4 |}$

Factor $3$ out of $3 (3 t) + 3 (- 4)$ .

$f' (t) = \frac{3 (3 t - 4)}{| 3 t - 4 |}$

The first derivative of $f (t)$ with respect to $t$ is $\frac{3 (3 t - 4)}{| 3 t - 4 |}$ .

$\frac{3 (3 t - 4)}{| 3 t - 4 |}$

Set the first derivative equal to $0$ then solve the equation $\frac{3 (3 t - 4)}{| 3 t - 4 |} = 0$ .

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Set the first derivative equal to $0$ .

$\frac{3 (3 t - 4)}{| 3 t - 4 |} = 0$

Set the numerator equal to zero.

$3 (3 t - 4) = 0$

Solve the equation for $t$ .

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Divide each term in $3 (3 t - 4) = 0$ by $3$ and simplify.

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Divide each term in $3 (3 t - 4) = 0$ by $3$ .

$\frac{3 (3 t - 4)}{3} = \frac{0}{3}$

Simplify the left side.

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Cancel the common factor of $3$ .

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Cancel the common factor.

$\frac{3 (3 t - 4)}{3} = \frac{0}{3}$

Divide $3 t - 4$ by $1$ .

$3 t - 4 = \frac{0}{3}$

Simplify the right side.

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Divide $0$ by $3$ .

$3 t - 4 = 0$

Add $4$ to both sides of the equation.

$3 t = 4$

Divide each term in $3 t = 4$ by $3$ and simplify.

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Divide each term in $3 t = 4$ by $3$ .

$\frac{3 t}{3} = \frac{4}{3}$

Simplify the left side.

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Cancel the common factor of $3$ .

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Cancel the common factor.

$\frac{3 t}{3} = \frac{4}{3}$

Divide $t$ by $1$ .

$t = \frac{4}{3}$

Find the values where the derivative is undefined.

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Set the denominator in $\frac{3 (3 t - 4)}{| 3 t - 4 |}$ equal to $0$ to find where the expression is undefined.

$| 3 t - 4 | = 0$

Solve for $t$ .

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Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$3 t - 4 = \pm 0$

Plus or minus $0$ is $0$ .

$3 t - 4 = 0$

Add $4$ to both sides of the equation.

$3 t = 4$

Divide each term in $3 t = 4$ by $3$ and simplify.

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Divide each term in $3 t = 4$ by $3$ .

$\frac{3 t}{3} = \frac{4}{3}$

Simplify the left side.

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Cancel the common factor of $3$ .

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Cancel the common factor.

$\frac{3 t}{3} = \frac{4}{3}$

Divide $t$ by $1$ .

$t = \frac{4}{3}$

Evaluate $| 3 t - 4 |$ at each $t$ value where the derivative is $0$ or undefined.

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Evaluate at $t = \frac{4}{3}$ .

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Substitute $\frac{4}{3}$ for $t$ .

$| 3 (\frac{4}{3}) - 4 |$

Simplify.

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Cancel the common factor of $3$ .

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Cancel the common factor.

$| 3 (\frac{4}{3}) - 4 |$

Rewrite the expression.

$| 4 - 4 |$

Subtract $4$ from $4$ .

$| 0 |$

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

$0$

List all of the points.

$(\frac{4}{3}, 0)$