Calculus Examples

$\frac{1}{2} t^{2} + 1$

Step 1

By the Sum Rule, the derivative of $\frac{1}{2} t^{2} + 1$ with respect to $t$ is $\frac{d}{d t} [\frac{1}{2} t^{2}] + \frac{d}{d t} [1]$ .

$\frac{d}{d t} [\frac{1}{2} t^{2}] + \frac{d}{d t} [1]$

Step 2

Evaluate $\frac{d}{d t} [\frac{1}{2} t^{2}]$ .

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Step 2.1

Since $\frac{1}{2}$ is constant with respect to $t$ , the derivative of $\frac{1}{2} t^{2}$ with respect to $t$ is $\frac{1}{2} \frac{d}{d t} [t^{2}]$ .

$\frac{1}{2} \frac{d}{d t} [t^{2}] + \frac{d}{d t} [1]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 2$ .

$\frac{1}{2} (2 t) + \frac{d}{d t} [1]$

Step 2.3

Combine $2$ and $\frac{1}{2}$ .

$\frac{2}{2} t + \frac{d}{d t} [1]$

Step 2.4

Combine $\frac{2}{2}$ and $t$ .

$\frac{2 t}{2} + \frac{d}{d t} [1]$

Step 2.5

Cancel the common factor of $2$ .

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Step 2.5.1

Cancel the common factor.

$\frac{2 t}{2} + \frac{d}{d t} [1]$

Step 2.5.2

Divide $t$ by $1$ .

$t + \frac{d}{d t} [1]$

Step 3

Differentiate using the Constant Rule.

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Step 3.1

Since $1$ is constant with respect to $t$ , the derivative of $1$ with respect to $t$ is $0$ .

$t + 0$

Step 3.2

Add $t$ and $0$ .

$t$