Calculus Examples

$g (t) = | 3 t - 4 |$

Step 1

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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Step 1.1

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

$\frac{d}{d u} [| u |] \frac{d}{d t} [3 t - 4]$

Step 1.2

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

$\frac{u}{| u |} \frac{d}{d t} [3 t - 4]$

Step 1.3

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

$\frac{3 t - 4}{| 3 t - 4 |} \frac{d}{d t} [3 t - 4]$

Step 2

Differentiate.

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Step 2.1

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

$\frac{3 t - 4}{| 3 t - 4 |} (\frac{d}{d t} [3 t] + \frac{d}{d t} [- 4])$

Step 2.2

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

$\frac{3 t - 4}{| 3 t - 4 |} (3 \frac{d}{d t} [t] + \frac{d}{d t} [- 4])$

Step 2.3

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

$\frac{3 t - 4}{| 3 t - 4 |} (3 \cdot 1 + \frac{d}{d t} [- 4])$

Step 2.4

Multiply $3$ by $1$ .

$\frac{3 t - 4}{| 3 t - 4 |} (3 + \frac{d}{d t} [- 4])$

Step 2.5

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

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

Step 2.6

Combine fractions.

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Step 2.6.1

Add $3$ and $0$ .

$\frac{3 t - 4}{| 3 t - 4 |} \cdot 3$

Step 2.6.2

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

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

Step 2.6.3

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

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Simplify each term.

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Step 3.2.1

Multiply $3$ by $3$ .

$\frac{9 t + 3 \cdot - 4}{| 3 t - 4 |}$

Step 3.2.2

Multiply $3$ by $- 4$ .

$\frac{9 t - 12}{| 3 t - 4 |}$

Step 3.3

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

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Step 3.3.1

Factor $3$ out of $9 t$ .

$\frac{3 (3 t) - 12}{| 3 t - 4 |}$

Step 3.3.2

Factor $3$ out of $- 12$ .

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

Step 3.3.3

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

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