Calculus Examples

$s = 7 t^{3} - 9 t^{2}$ , $t = 4$

Step 1

By the Sum Rule, the derivative of $7 t^{3} - 9 t^{2}$ with respect to $t$ is $\frac{d}{d t} [7 t^{3}] + \frac{d}{d t} [- 9 t^{2}]$ .

$\frac{d}{d t} [7 t^{3}] + \frac{d}{d t} [- 9 t^{2}]$

Step 2

Evaluate $\frac{d}{d t} [7 t^{3}]$ .

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Step 2.1

Since $7$ is constant with respect to $t$ , the derivative of $7 t^{3}$ with respect to $t$ is $7 \frac{d}{d t} [t^{3}]$ .

$7 \frac{d}{d t} [t^{3}] + \frac{d}{d t} [- 9 t^{2}]$

Step 2.2

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

$7 (3 t^{2}) + \frac{d}{d t} [- 9 t^{2}]$

Step 2.3

Multiply $3$ by $7$ .

$21 t^{2} + \frac{d}{d t} [- 9 t^{2}]$

Step 3

Evaluate $\frac{d}{d t} [- 9 t^{2}]$ .

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Step 3.1

Since $- 9$ is constant with respect to $t$ , the derivative of $- 9 t^{2}$ with respect to $t$ is $- 9 \frac{d}{d t} [t^{2}]$ .

$21 t^{2} - 9 \frac{d}{d t} [t^{2}]$

Step 3.2

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

$21 t^{2} - 9 (2 t)$

Step 3.3

Multiply $2$ by $- 9$ .

$21 t^{2} - 18 t$

Step 4

Evaluate the derivative at $t = 4$ .

$21 {(4)}^{2} - 18 \cdot 4$

Step 5

Simplify.

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Step 5.1

Simplify each term.

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Step 5.1.1

Raise $4$ to the power of $2$ .

$21 \cdot 16 - 18 \cdot 4$

Step 5.1.2

Multiply $21$ by $16$ .

$336 - 18 \cdot 4$

Step 5.1.3

Multiply $- 18$ by $4$ .

$336 - 72$

Step 5.2

Subtract $72$ from $336$ .

$264$