Calculus Examples

$s = 120 t - 5 t^{2}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d t} (s) = \frac{d}{d t} (120 t - 5 t^{2})$

Step 2

The derivative of $s$ with respect to $t$ is $s'$ .

$s'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $120 t - 5 t^{2}$ with respect to $t$ is $\frac{d}{d t} [120 t] + \frac{d}{d t} [- 5 t^{2}]$ .

$\frac{d}{d t} [120 t] + \frac{d}{d t} [- 5 t^{2}]$

Step 3.2

Evaluate $\frac{d}{d t} [120 t]$ .

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Step 3.2.1

Since $120$ is constant with respect to $t$ , the derivative of $120 t$ with respect to $t$ is $120 \frac{d}{d t} [t]$ .

$120 \frac{d}{d t} [t] + \frac{d}{d t} [- 5 t^{2}]$

Step 3.2.2

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

$120 \cdot 1 + \frac{d}{d t} [- 5 t^{2}]$

Step 3.2.3

Multiply $120$ by $1$ .

$120 + \frac{d}{d t} [- 5 t^{2}]$

Step 3.3

Evaluate $\frac{d}{d t} [- 5 t^{2}]$ .

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Step 3.3.1

Since $- 5$ is constant with respect to $t$ , the derivative of $- 5 t^{2}$ with respect to $t$ is $- 5 \frac{d}{d t} [t^{2}]$ .

$120 - 5 \frac{d}{d t} [t^{2}]$

Step 3.3.2

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

$120 - 5 (2 t)$

Step 3.3.3

Multiply $2$ by $- 5$ .

$120 - 10 t$

Step 3.4

Reorder terms.

$- 10 t + 120$

Step 4

Reform the equation by setting the left side equal to the right side.

$s' = - 10 t + 120$

Step 5

Replace $s'$ with $\frac{d s}{d t}$ .

$\frac{d s}{d t} = - 10 t + 120$