Calculus Examples

$x = 2 t^{2} + t$ , $t = 3$

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Multiply $2$ by $2$ .

$4 t + \frac{d}{d t} [t]$

Step 3

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

$4 t + 1$

Step 4

Evaluate the derivative at $t = 3$ .

$4 (3) + 1$

Step 5

Simplify.

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Step 5.1

Multiply $4$ by $3$ .

$12 + 1$

Step 5.2

Add $12$ and $1$ .

$13$