Calculus Examples

$s = 4 t^{2} - 5 t$ , $t = 2$

Step 1

By the Sum Rule, the derivative of $4 t^{2} - 5 t$ with respect to $t$ is $\frac{d}{d t} [4 t^{2}] + \frac{d}{d t} [- 5 t]$ .

$\frac{d}{d t} [4 t^{2}] + \frac{d}{d t} [- 5 t]$

Step 2

Evaluate $\frac{d}{d t} [4 t^{2}]$ .

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Step 2.1

Since $4$ is constant with respect to $t$ , the derivative of $4 t^{2}$ with respect to $t$ is $4 \frac{d}{d t} [t^{2}]$ .

$4 \frac{d}{d t} [t^{2}] + \frac{d}{d t} [- 5 t]$

Step 2.2

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

$4 (2 t) + \frac{d}{d t} [- 5 t]$

Step 2.3

Multiply $2$ by $4$ .

$8 t + \frac{d}{d t} [- 5 t]$

Step 3

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

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Step 3.1

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

$8 t - 5 \frac{d}{d t} [t]$

Step 3.2

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

$8 t - 5 \cdot 1$

Step 3.3

Multiply $- 5$ by $1$ .

$8 t - 5$

Step 4

Evaluate the derivative at $t = 2$ .

$8 (2) - 5$

Step 5

Simplify.

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Step 5.1

Multiply $8$ by $2$ .

$16 - 5$

Step 5.2

Subtract $5$ from $16$ .

$11$