Calculus Examples

$\frac{1}{3} t^{3} - t$

Step 1

By the Sum Rule, the derivative of $\frac{1}{3} t^{3} - t$ with respect to $t$ is $\frac{d}{d t} [\frac{1}{3} t^{3}] + \frac{d}{d t} [- t]$ .

$\frac{d}{d t} [\frac{1}{3} t^{3}] + \frac{d}{d t} [- t]$

Step 2

Evaluate $\frac{d}{d t} [\frac{1}{3} t^{3}]$ .

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Step 2.1

Since $\frac{1}{3}$ is constant with respect to $t$ , the derivative of $\frac{1}{3} t^{3}$ with respect to $t$ is $\frac{1}{3} \frac{d}{d t} [t^{3}]$ .

$\frac{1}{3} \frac{d}{d t} [t^{3}] + \frac{d}{d t} [- t]$

Step 2.2

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

$\frac{1}{3} (3 t^{2}) + \frac{d}{d t} [- t]$

Step 2.3

Combine $3$ and $\frac{1}{3}$ .

$\frac{3}{3} t^{2} + \frac{d}{d t} [- t]$

Step 2.4

Combine $\frac{3}{3}$ and $t^{2}$ .

$\frac{3 t^{2}}{3} + \frac{d}{d t} [- t]$

Step 2.5

Cancel the common factor of $3$ .

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Step 2.5.1

Cancel the common factor.

$\frac{3 t^{2}}{3} + \frac{d}{d t} [- t]$

Step 2.5.2

Divide $t^{2}$ by $1$ .

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

Step 3

Evaluate $\frac{d}{d t} [- t]$ .

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Step 3.1

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

$t^{2} - \frac{d}{d t} [t]$

Step 3.2

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

$t^{2} - 1 \cdot 1$

Step 3.3

Multiply $- 1$ by $1$ .

$t^{2} - 1$