Calculus Examples

$s = 8 t + 17$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d t} (s) = \frac{d}{d t} (8 t + 17)$

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 $8 t + 17$ with respect to $t$ is $\frac{d}{d t} [8 t] + \frac{d}{d t} [17]$ .

$\frac{d}{d t} [8 t] + \frac{d}{d t} [17]$

Step 3.2

Evaluate $\frac{d}{d t} [8 t]$ .

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Step 3.2.1

Since $8$ is constant with respect to $t$ , the derivative of $8 t$ with respect to $t$ is $8 \frac{d}{d t} [t]$ .

$8 \frac{d}{d t} [t] + \frac{d}{d t} [17]$

Step 3.2.2

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

$8 \cdot 1 + \frac{d}{d t} [17]$

Step 3.2.3

Multiply $8$ by $1$ .

$8 + \frac{d}{d t} [17]$

Step 3.3

Differentiate using the Constant Rule.

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Step 3.3.1

Since $17$ is constant with respect to $t$ , the derivative of $17$ with respect to $t$ is $0$ .

$8 + 0$

Step 3.3.2

Add $8$ and $0$ .

$8$

Step 4

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

$s' = 8$

Step 5

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

$\frac{d s}{d t} = 8$