Calculus Examples

$s = 5 t^{4} - 4 t^{3}$ , $t = - 1$

Step 1

By the Sum Rule, the derivative of $5 t^{4} - 4 t^{3}$ with respect to $t$ is $\frac{d}{d t} [5 t^{4}] + \frac{d}{d t} [- 4 t^{3}]$ .

$\frac{d}{d t} [5 t^{4}] + \frac{d}{d t} [- 4 t^{3}]$

Step 2

Evaluate $\frac{d}{d t} [5 t^{4}]$ .

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Step 2.1

Since $5$ is constant with respect to $t$ , the derivative of $5 t^{4}$ with respect to $t$ is $5 \frac{d}{d t} [t^{4}]$ .

$5 \frac{d}{d t} [t^{4}] + \frac{d}{d t} [- 4 t^{3}]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 4$ .

$5 (4 t^{3}) + \frac{d}{d t} [- 4 t^{3}]$

Step 2.3

Multiply $4$ by $5$ .

$20 t^{3} + \frac{d}{d t} [- 4 t^{3}]$

Step 3

Evaluate $\frac{d}{d t} [- 4 t^{3}]$ .

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Step 3.1

Since $- 4$ is constant with respect to $t$ , the derivative of $- 4 t^{3}$ with respect to $t$ is $- 4 \frac{d}{d t} [t^{3}]$ .

$20 t^{3} - 4 \frac{d}{d t} [t^{3}]$

Step 3.2

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

$20 t^{3} - 4 (3 t^{2})$

Step 3.3

Multiply $3$ by $- 4$ .

$20 t^{3} - 12 t^{2}$

Step 4

Evaluate the derivative at $t = - 1$ .

$20 {(- 1)}^{3} - 12 {(- 1)}^{2}$

Step 5

Simplify.

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Step 5.1

Simplify each term.

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Step 5.1.1

Raise $- 1$ to the power of $3$ .

$20 \cdot - 1 - 12 {(- 1)}^{2}$

Step 5.1.2

Multiply $20$ by $- 1$ .

$- 20 - 12 {(- 1)}^{2}$

Step 5.1.3

Raise $- 1$ to the power of $2$ .

$- 20 - 12 \cdot 1$

Step 5.1.4

Multiply $- 12$ by $1$ .

$- 20 - 12$

Step 5.2

Subtract $12$ from $- 20$ .

$- 32$