Calculus Examples

$s (t) = 5 t^{2} + 15 t$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $5 t^{2} + 15 t$ with respect to $t$ is $\frac{d}{d t} [5 t^{2}] + \frac{d}{d t} [15 t]$ .

$\frac{d}{d t} [5 t^{2}] + \frac{d}{d t} [15 t]$

Step 1.2

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

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Step 1.2.1

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

$5 \frac{d}{d t} [t^{2}] + \frac{d}{d t} [15 t]$

Step 1.2.2

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

$5 (2 t) + \frac{d}{d t} [15 t]$

Step 1.2.3

Multiply $2$ by $5$ .

$10 t + \frac{d}{d t} [15 t]$

Step 1.3

Evaluate $\frac{d}{d t} [15 t]$ .

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Step 1.3.1

Since $15$ is constant with respect to $t$ , the derivative of $15 t$ with respect to $t$ is $15 \frac{d}{d t} [t]$ .

$10 t + 15 \frac{d}{d t} [t]$

Step 1.3.2

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

$10 t + 15 \cdot 1$

Step 1.3.3

Multiply $15$ by $1$ .

$f' (t) = 10 t + 15$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $10 t + 15$ with respect to $t$ is $\frac{d}{d t} [10 t] + \frac{d}{d t} [15]$ .

$\frac{d}{d t} [10 t] + \frac{d}{d t} [15]$

Step 2.2

Evaluate $\frac{d}{d t} [10 t]$ .

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Step 2.2.1

Since $10$ is constant with respect to $t$ , the derivative of $10 t$ with respect to $t$ is $10 \frac{d}{d t} [t]$ .

$10 \frac{d}{d t} [t] + \frac{d}{d t} [15]$

Step 2.2.2

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

$10 \cdot 1 + \frac{d}{d t} [15]$

Step 2.2.3

Multiply $10$ by $1$ .

$10 + \frac{d}{d t} [15]$

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

Since $15$ is constant with respect to $t$ , the derivative of $15$ with respect to $t$ is $0$ .

$10 + 0$

Step 2.3.2

Add $10$ and $0$ .

$f'' (t) = 10$

Step 3

Since $10$ is constant with respect to $t$ , the derivative of $10$ with respect to $t$ is $0$ .

$f''' (t) = 0$

Step 4

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

$f^{4} (t) = 0$

Step 5

The fourth derivative of $s (t)$ with respect to $t$ is $0$ .

$0$