Calculus Examples

$h (s) = s^{3} (s^{2} - 4 s + 4)$

Step 1

Find the first derivative.

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Step 1.1

Differentiate using the Product Rule which states that $\frac{d}{d s} [f (s) g (s)]$ is $f (s) \frac{d}{d s} [g (s)] + g (s) \frac{d}{d s} [f (s)]$ where $f (s) = s^{3}$ and $g (s) = s^{2} - 4 s + 4$ .

$s^{3} \frac{d}{d s} [s^{2} - 4 s + 4] + (s^{2} - 4 s + 4) \frac{d}{d s} [s^{3}]$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $s^{2} - 4 s + 4$ with respect to $s$ is $\frac{d}{d s} [s^{2}] + \frac{d}{d s} [- 4 s] + \frac{d}{d s} [4]$ .

$s^{3} (\frac{d}{d s} [s^{2}] + \frac{d}{d s} [- 4 s] + \frac{d}{d s} [4]) + (s^{2} - 4 s + 4) \frac{d}{d s} [s^{3}]$

Step 1.2.2

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

$s^{3} (2 s + \frac{d}{d s} [- 4 s] + \frac{d}{d s} [4]) + (s^{2} - 4 s + 4) \frac{d}{d s} [s^{3}]$

Step 1.2.3

Since $- 4$ is constant with respect to $s$ , the derivative of $- 4 s$ with respect to $s$ is $- 4 \frac{d}{d s} [s]$ .

$s^{3} (2 s - 4 \frac{d}{d s} [s] + \frac{d}{d s} [4]) + (s^{2} - 4 s + 4) \frac{d}{d s} [s^{3}]$

Step 1.2.4

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

$s^{3} (2 s - 4 \cdot 1 + \frac{d}{d s} [4]) + (s^{2} - 4 s + 4) \frac{d}{d s} [s^{3}]$

Step 1.2.5

Multiply $- 4$ by $1$ .

$s^{3} (2 s - 4 + \frac{d}{d s} [4]) + (s^{2} - 4 s + 4) \frac{d}{d s} [s^{3}]$

Step 1.2.6

Since $4$ is constant with respect to $s$ , the derivative of $4$ with respect to $s$ is $0$ .

$s^{3} (2 s - 4 + 0) + (s^{2} - 4 s + 4) \frac{d}{d s} [s^{3}]$

Step 1.2.7

Add $2 s - 4$ and $0$ .

$s^{3} (2 s - 4) + (s^{2} - 4 s + 4) \frac{d}{d s} [s^{3}]$

Step 1.2.8

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

$s^{3} (2 s - 4) + (s^{2} - 4 s + 4) (3 s^{2})$

Step 1.2.9

Move $3$ to the left of $s^{2} - 4 s + 4$ .

$s^{3} (2 s - 4) + 3 \cdot (s^{2} - 4 s + 4) s^{2}$

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

$s^{3} (2 s) + s^{3} \cdot - 4 + 3 (s^{2} - 4 s + 4) s^{2}$

Step 1.3.2

Apply the distributive property.

$s^{3} (2 s) + s^{3} \cdot - 4 + (3 s^{2} + 3 (- 4 s) + 3 \cdot 4) s^{2}$

Step 1.3.3

Apply the distributive property.

$s^{3} (2 s) + s^{3} \cdot - 4 + 3 s^{2} s^{2} + 3 (- 4 s) s^{2} + 3 \cdot 4 s^{2}$

Step 1.3.4

Combine terms.

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Step 1.3.4.1

Multiply $s^{3}$ by $s$ by adding the exponents.

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Step 1.3.4.1.1

Move $s$ .

$s \cdot s^{3} \cdot 2 + s^{3} \cdot - 4 + 3 s^{2} s^{2} + 3 (- 4 s) s^{2} + 3 \cdot 4 s^{2}$

Step 1.3.4.1.2

Multiply $s$ by $s^{3}$ .

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Step 1.3.4.1.2.1

Raise $s$ to the power of $1$ .

$s^{1} s^{3} \cdot 2 + s^{3} \cdot - 4 + 3 s^{2} s^{2} + 3 (- 4 s) s^{2} + 3 \cdot 4 s^{2}$

Step 1.3.4.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$s^{1 + 3} \cdot 2 + s^{3} \cdot - 4 + 3 s^{2} s^{2} + 3 (- 4 s) s^{2} + 3 \cdot 4 s^{2}$

Step 1.3.4.1.3

Add $1$ and $3$ .

$s^{4} \cdot 2 + s^{3} \cdot - 4 + 3 s^{2} s^{2} + 3 (- 4 s) s^{2} + 3 \cdot 4 s^{2}$

Step 1.3.4.2

Move $2$ to the left of $s^{4}$ .

$2 \cdot s^{4} + s^{3} \cdot - 4 + 3 s^{2} s^{2} + 3 (- 4 s) s^{2} + 3 \cdot 4 s^{2}$

Step 1.3.4.3

Move $- 4$ to the left of $s^{3}$ .

$2 s^{4} - 4 \cdot s^{3} + 3 s^{2} s^{2} + 3 (- 4 s) s^{2} + 3 \cdot 4 s^{2}$

Step 1.3.4.4

Multiply $s^{2}$ by $s^{2}$ by adding the exponents.

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Step 1.3.4.4.1

Move $s^{2}$ .

$2 s^{4} - 4 s^{3} + 3 (s^{2} s^{2}) + 3 (- 4 s) s^{2} + 3 \cdot 4 s^{2}$

Step 1.3.4.4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 s^{4} - 4 s^{3} + 3 s^{2 + 2} + 3 (- 4 s) s^{2} + 3 \cdot 4 s^{2}$

Step 1.3.4.4.3

Add $2$ and $2$ .

$2 s^{4} - 4 s^{3} + 3 s^{4} + 3 (- 4 s) s^{2} + 3 \cdot 4 s^{2}$

Step 1.3.4.5

Multiply $- 4$ by $3$ .

$2 s^{4} - 4 s^{3} + 3 s^{4} - 12 s \cdot s^{2} + 3 \cdot 4 s^{2}$

Step 1.3.4.6

Multiply $s$ by $s^{2}$ by adding the exponents.

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Step 1.3.4.6.1

Move $s^{2}$ .

$2 s^{4} - 4 s^{3} + 3 s^{4} - 12 (s^{2} s) + 3 \cdot 4 s^{2}$

Step 1.3.4.6.2

Multiply $s^{2}$ by $s$ .

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Step 1.3.4.6.2.1

Raise $s$ to the power of $1$ .

$2 s^{4} - 4 s^{3} + 3 s^{4} - 12 (s^{2} s^{1}) + 3 \cdot 4 s^{2}$

Step 1.3.4.6.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 s^{4} - 4 s^{3} + 3 s^{4} - 12 s^{2 + 1} + 3 \cdot 4 s^{2}$

Step 1.3.4.6.3

Add $2$ and $1$ .

$2 s^{4} - 4 s^{3} + 3 s^{4} - 12 s^{3} + 3 \cdot 4 s^{2}$

Step 1.3.4.7

Multiply $3$ by $4$ .

$2 s^{4} - 4 s^{3} + 3 s^{4} - 12 s^{3} + 12 s^{2}$

Step 1.3.4.8

Add $2 s^{4}$ and $3 s^{4}$ .

$5 s^{4} - 4 s^{3} - 12 s^{3} + 12 s^{2}$

Step 1.3.4.9

Subtract $12 s^{3}$ from $- 4 s^{3}$ .

$f' (s) = 5 s^{4} - 16 s^{3} + 12 s^{2}$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $5 s^{4} - 16 s^{3} + 12 s^{2}$ with respect to $s$ is $\frac{d}{d s} [5 s^{4}] + \frac{d}{d s} [- 16 s^{3}] + \frac{d}{d s} [12 s^{2}]$ .

$\frac{d}{d s} [5 s^{4}] + \frac{d}{d s} [- 16 s^{3}] + \frac{d}{d s} [12 s^{2}]$

Step 2.2

Evaluate $\frac{d}{d s} [5 s^{4}]$ .

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Step 2.2.1

Since $5$ is constant with respect to $s$ , the derivative of $5 s^{4}$ with respect to $s$ is $5 \frac{d}{d s} [s^{4}]$ .

$5 \frac{d}{d s} [s^{4}] + \frac{d}{d s} [- 16 s^{3}] + \frac{d}{d s} [12 s^{2}]$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d s} [s^{n}]$ is $n s^{n - 1}$ where $n = 4$ .

$5 (4 s^{3}) + \frac{d}{d s} [- 16 s^{3}] + \frac{d}{d s} [12 s^{2}]$

Step 2.2.3

Multiply $4$ by $5$ .

$20 s^{3} + \frac{d}{d s} [- 16 s^{3}] + \frac{d}{d s} [12 s^{2}]$

Step 2.3

Evaluate $\frac{d}{d s} [- 16 s^{3}]$ .

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Step 2.3.1

Since $- 16$ is constant with respect to $s$ , the derivative of $- 16 s^{3}$ with respect to $s$ is $- 16 \frac{d}{d s} [s^{3}]$ .

$20 s^{3} - 16 \frac{d}{d s} [s^{3}] + \frac{d}{d s} [12 s^{2}]$

Step 2.3.2

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

$20 s^{3} - 16 (3 s^{2}) + \frac{d}{d s} [12 s^{2}]$

Step 2.3.3

Multiply $3$ by $- 16$ .

$20 s^{3} - 48 s^{2} + \frac{d}{d s} [12 s^{2}]$

Step 2.4

Evaluate $\frac{d}{d s} [12 s^{2}]$ .

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Step 2.4.1

Since $12$ is constant with respect to $s$ , the derivative of $12 s^{2}$ with respect to $s$ is $12 \frac{d}{d s} [s^{2}]$ .

$20 s^{3} - 48 s^{2} + 12 \frac{d}{d s} [s^{2}]$

Step 2.4.2

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

$20 s^{3} - 48 s^{2} + 12 (2 s)$

Step 2.4.3

Multiply $2$ by $12$ .

$f'' (s) = 20 s^{3} - 48 s^{2} + 24 s$

Step 3

The second derivative of $h (s)$ with respect to $s$ is $20 s^{3} - 48 s^{2} + 24 s$ .

$20 s^{3} - 48 s^{2} + 24 s$