Calculus Examples

$f (x) = \frac{1}{4} \cdot (t^{4} + 8)$

Step 1

Since $\frac{1}{4}$ is constant with respect to $t$ , the derivative of $\frac{1}{4} (t^{4} + 8)$ with respect to $t$ is $\frac{1}{4} \frac{d}{d t} [t^{4} + 8]$ .

$\frac{1}{4} \frac{d}{d t} [t^{4} + 8]$

Step 2

By the Sum Rule, the derivative of $t^{4} + 8$ with respect to $t$ is $\frac{d}{d t} [t^{4}] + \frac{d}{d t} [8]$ .

$\frac{1}{4} (\frac{d}{d t} [t^{4}] + \frac{d}{d t} [8])$

Step 3

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

$\frac{1}{4} (4 t^{3} + \frac{d}{d t} [8])$

Step 4

Since $8$ is constant with respect to $t$ , the derivative of $8$ with respect to $t$ is $0$ .

$\frac{1}{4} (4 t^{3} + 0)$

Step 5

Simplify terms.

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Step 5.1

Add $4 t^{3}$ and $0$ .

$\frac{1}{4} (4 t^{3})$

Step 5.2

Combine $4$ and $\frac{1}{4}$ .

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

Step 5.3

Combine $\frac{4}{4}$ and $t^{3}$ .

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

Step 5.4

Cancel the common factor of $4$ .

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Step 5.4.1

Cancel the common factor.

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

Step 5.4.2

Divide $t^{3}$ by $1$ .

$t^{3}$