Calculus Examples

$s (t) = 2 t^{2} + 2 t$ , $t = 4$

Step 1

Find the derivative.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

Multiply $2$ by $2$ .

$4 t + \frac{d}{d t} [2 t]$

Step 1.3

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

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Step 1.3.1

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

$4 t + 2 \frac{d}{d t} [t]$

Step 1.3.2

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

$4 t + 2 \cdot 1$

Step 1.3.3

Multiply $2$ by $1$ .

$4 t + 2$

Step 2

Replace the variable $t$ with $4$ in the expression.

$s' (4) = 4 (4) + 2$

Step 3

Multiply $4$ by $4$ .

$s' (4) = 16 + 2$

Step 4

Add $16$ and $2$ .

$s' (4) = 18$