Calculus Examples

$S (t) = \frac{1}{3} t^{3} - 3.5 t^{2} + 10 t$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $\frac{1}{3} t^{3} - 3.5 t^{2} + 10 t$ with respect to $t$ is $\frac{d}{d t} [\frac{1}{3} t^{3}] + \frac{d}{d t} [- 3.5 t^{2}] + \frac{d}{d t} [10 t]$ .

$\frac{d}{d t} [\frac{1}{3} t^{3}] + \frac{d}{d t} [- 3.5 t^{2}] + \frac{d}{d t} [10 t]$

Step 1.2

Evaluate $\frac{d}{d t} [\frac{1}{3} t^{3}]$ .

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Step 1.2.1

Since $\frac{1}{3}$ is constant with respect to $t$ , the derivative of $\frac{1}{3} t^{3}$ with respect to $t$ is $\frac{1}{3} \frac{d}{d t} [t^{3}]$ .

$\frac{1}{3} \frac{d}{d t} [t^{3}] + \frac{d}{d t} [- 3.5 t^{2}] + \frac{d}{d t} [10 t]$

Step 1.2.2

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

$\frac{1}{3} (3 t^{2}) + \frac{d}{d t} [- 3.5 t^{2}] + \frac{d}{d t} [10 t]$

Step 1.2.3

Combine $3$ and $\frac{1}{3}$ .

$\frac{3}{3} t^{2} + \frac{d}{d t} [- 3.5 t^{2}] + \frac{d}{d t} [10 t]$

Step 1.2.4

Combine $\frac{3}{3}$ and $t^{2}$ .

$\frac{3 t^{2}}{3} + \frac{d}{d t} [- 3.5 t^{2}] + \frac{d}{d t} [10 t]$

Step 1.2.5

Cancel the common factor of $3$ .

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Step 1.2.5.1

Cancel the common factor.

$\frac{3 t^{2}}{3} + \frac{d}{d t} [- 3.5 t^{2}] + \frac{d}{d t} [10 t]$

Step 1.2.5.2

Divide $t^{2}$ by $1$ .

$t^{2} + \frac{d}{d t} [- 3.5 t^{2}] + \frac{d}{d t} [10 t]$

Step 1.3

Evaluate $\frac{d}{d t} [- 3.5 t^{2}]$ .

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Step 1.3.1

Since $- 3.5$ is constant with respect to $t$ , the derivative of $- 3.5 t^{2}$ with respect to $t$ is $- 3.5 \frac{d}{d t} [t^{2}]$ .

$t^{2} - 3.5 \frac{d}{d t} [t^{2}] + \frac{d}{d t} [10 t]$

Step 1.3.2

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

$t^{2} - 3.5 (2 t) + \frac{d}{d t} [10 t]$

Step 1.3.3

Multiply $2$ by $- 3.5$ .

$t^{2} - 7 t + \frac{d}{d t} [10 t]$

Step 1.4

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

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Step 1.4.1

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

$t^{2} - 7 t + 10 \frac{d}{d t} [t]$

Step 1.4.2

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

$t^{2} - 7 t + 10 \cdot 1$

Step 1.4.3

Multiply $10$ by $1$ .

$f' (t) = t^{2} - 7 t + 10$

Step 2

Find the second derivative.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $t^{2} - 7 t + 10$ with respect to $t$ is $\frac{d}{d t} [t^{2}] + \frac{d}{d t} [- 7 t] + \frac{d}{d t} [10]$ .

$\frac{d}{d t} [t^{2}] + \frac{d}{d t} [- 7 t] + \frac{d}{d t} [10]$

Step 2.1.2

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

$2 t + \frac{d}{d t} [- 7 t] + \frac{d}{d t} [10]$

Step 2.2

Evaluate $\frac{d}{d t} [- 7 t]$ .

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Step 2.2.1

Since $- 7$ is constant with respect to $t$ , the derivative of $- 7 t$ with respect to $t$ is $- 7 \frac{d}{d t} [t]$ .

$2 t - 7 \frac{d}{d t} [t] + \frac{d}{d t} [10]$

Step 2.2.2

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

$2 t - 7 \cdot 1 + \frac{d}{d t} [10]$

Step 2.2.3

Multiply $- 7$ by $1$ .

$2 t - 7 + \frac{d}{d t} [10]$

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

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

$2 t - 7 + 0$

Step 2.3.2

Add $2 t - 7$ and $0$ .

$f'' (t) = 2 t - 7$

Step 3

The second derivative of $S (t)$ with respect to $t$ is $2 t - 7$ .

$2 t - 7$