Calculus Examples

$\sqrt{72 t^{6}}$

Step 1

Rewrite $\frac{d}{d t} [\sqrt{72 t^{6}}]$ as $\frac{d}{d t} [6 t^{3} \sqrt{2}]$ .

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Step 1.1

Rewrite $72 t^{6}$ as ${(6 t^{3})}^{2} \cdot 2$ .

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Step 1.1.1

Factor $36$ out of $72$ .

$\frac{d}{d t} [\sqrt{36 (2) t^{6}}]$

Step 1.1.2

Rewrite $36$ as $6^{2}$ .

$\frac{d}{d t} [\sqrt{6^{2} \cdot 2 t^{6}}]$

Step 1.1.3

Rewrite $t^{6}$ as ${(t^{3})}^{2}$ .

$\frac{d}{d t} [\sqrt{6^{2} \cdot 2 {(t^{3})}^{2}}]$

Step 1.1.4

Move $2$ .

$\frac{d}{d t} [\sqrt{6^{2} {(t^{3})}^{2} \cdot 2}]$

Step 1.1.5

Rewrite $6^{2} {(t^{3})}^{2}$ as ${(6 t^{3})}^{2}$ .

$\frac{d}{d t} [\sqrt{{(6 t^{3})}^{2} \cdot 2}]$

Step 1.2

Pull terms out from under the radical.

$\frac{d}{d t} [6 t^{3} \sqrt{2}]$

Step 2

Since $6 \sqrt{2}$ is constant with respect to $t$ , the derivative of $6 t^{3} \sqrt{2}$ with respect to $t$ is $6 \sqrt{2} \frac{d}{d t} [t^{3}]$ .

$6 \sqrt{2} \frac{d}{d t} [t^{3}]$

Step 3

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

$6 \sqrt{2} (3 t^{2})$

Step 4

Simplify the expression.

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Step 4.1

Multiply $3$ by $6$ .

$18 \sqrt{2} t^{2}$

Step 4.2

Reorder the factors of $18 \sqrt{2} t^{2}$ .

$18 t^{2} \sqrt{2}$