Calculus Examples

${(12 t^{2})}^{2}$

Step 1

Simplify with factoring out.

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Step 1.1

Factor $12$ out of $12 t^{2}$ .

$\frac{d}{d t} [{(12 (t^{2}))}^{2}]$

Step 1.2

Simplify the expression.

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Step 1.2.1

Apply the product rule to $12 (t^{2})$ .

$\frac{d}{d t} [12^{2} {(t^{2})}^{2}]$

Step 1.2.2

Raise $12$ to the power of $2$ .

$\frac{d}{d t} [144 {(t^{2})}^{2}]$

Step 1.2.3

Multiply the exponents in ${(t^{2})}^{2}$ .

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Step 1.2.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{d}{d t} [144 t^{2 \cdot 2}]$

Step 1.2.3.2

Multiply $2$ by $2$ .

$\frac{d}{d t} [144 t^{4}]$

Step 2

Since $144$ is constant with respect to $t$ , the derivative of $144 t^{4}$ with respect to $t$ is $144 \frac{d}{d t} [t^{4}]$ .

$144 \frac{d}{d t} [t^{4}]$

Step 3

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 4$ .

$144 (4 t^{3})$

Step 4

Multiply $4$ by $144$ .

$576 t^{3}$