Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

Differentiate.

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Step 1.1.1

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

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

Step 1.1.2

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

$2 t + \frac{d}{d t} [2 t]$

Step 1.2

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

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Step 1.2.1

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

$2 t + 2 \frac{d}{d t} [t]$

Step 1.2.2

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

$2 t + 2 \cdot 1$

Step 1.2.3

Multiply $2$ by $1$ .

$f' (t) = 2 t + 2$

Step 2

Find the second derivative.

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Step 2.1

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

$\frac{d}{d t} [2 t] + \frac{d}{d t} [2]$

Step 2.2

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

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Step 2.2.1

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

$2 \frac{d}{d t} [t] + \frac{d}{d t} [2]$

Step 2.2.2

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

$2 \cdot 1 + \frac{d}{d t} [2]$

Step 2.2.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d t} [2]$

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

Since $2$ is constant with respect to $t$ , the derivative of $2$ with respect to $t$ is $0$ .

$2 + 0$

Step 2.3.2

Add $2$ and $0$ .

$f'' (t) = 2$

Step 3

Since $2$ is constant with respect to $t$ , the derivative of $2$ 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 $0$ .

$0$