Calculus Examples

$s = 5 t^{3} - 2 t^{2}$ , $t = 3$

Step 1

By the Sum Rule, the derivative of $5 t^{3} - 2 t^{2}$ with respect to $t$ is $\frac{d}{d t} [5 t^{3}] + \frac{d}{d t} [- 2 t^{2}]$ .

$\frac{d}{d t} [5 t^{3}] + \frac{d}{d t} [- 2 t^{2}]$

Step 2

Evaluate $\frac{d}{d t} [5 t^{3}]$ .

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Step 2.1

Since $5$ is constant with respect to $t$ , the derivative of $5 t^{3}$ with respect to $t$ is $5 \frac{d}{d t} [t^{3}]$ .

$5 \frac{d}{d t} [t^{3}] + \frac{d}{d t} [- 2 t^{2}]$

Step 2.2

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

$5 (3 t^{2}) + \frac{d}{d t} [- 2 t^{2}]$

Step 2.3

Multiply $3$ by $5$ .

$15 t^{2} + \frac{d}{d t} [- 2 t^{2}]$

Step 3

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

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Step 3.1

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

$15 t^{2} - 2 \frac{d}{d t} [t^{2}]$

Step 3.2

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

$15 t^{2} - 2 (2 t)$

Step 3.3

Multiply $2$ by $- 2$ .

$15 t^{2} - 4 t$

Step 4

Evaluate the derivative at $t = 3$ .

$15 {(3)}^{2} - 4 \cdot 3$

Step 5

Simplify.

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Step 5.1

Simplify each term.

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Step 5.1.1

Raise $3$ to the power of $2$ .

$15 \cdot 9 - 4 \cdot 3$

Step 5.1.2

Multiply $15$ by $9$ .

$135 - 4 \cdot 3$

Step 5.1.3

Multiply $- 4$ by $3$ .

$135 - 12$

Step 5.2

Subtract $12$ from $135$ .

$123$