Calculus Examples

$\frac{8 (4 - 4 h + h^{2}) - 32 + 5}{h}$

Step 1

Add $- 32$ and $5$ .

$\frac{d}{d h} [\frac{8 (4 - 4 h + h^{2}) - 27}{h}]$

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d h} [\frac{f (h)}{g (h)}]$ is $\frac{g (h) \frac{d}{d h} [f (h)] - f (h) \frac{d}{d h} [g (h)]}{{g (h)}^{2}}$ where $f (h) = 8 (4 - 4 h + h^{2}) - 27$ and $g (h) = h$ .

$\frac{h \frac{d}{d h} [8 (4 - 4 h + h^{2}) - 27] - (8 (4 - 4 h + h^{2}) - 27) \frac{d}{d h} [h]}{h^{2}}$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $8 (4 - 4 h + h^{2}) - 27$ with respect to $h$ is $\frac{d}{d h} [8 (4 - 4 h + h^{2})] + \frac{d}{d h} [- 27]$ .

$\frac{h (\frac{d}{d h} [8 (4 - 4 h + h^{2})] + \frac{d}{d h} [- 27]) - (8 (4 - 4 h + h^{2}) - 27) \frac{d}{d h} [h]}{h^{2}}$

Step 3.2

Since $8$ is constant with respect to $h$ , the derivative of $8 (4 - 4 h + h^{2})$ with respect to $h$ is $8 \frac{d}{d h} [4 - 4 h + h^{2}]$ .

$\frac{h (8 \frac{d}{d h} [4 - 4 h + h^{2}] + \frac{d}{d h} [- 27]) - (8 (4 - 4 h + h^{2}) - 27) \frac{d}{d h} [h]}{h^{2}}$

Step 3.3

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

$\frac{h (8 (\frac{d}{d h} [4] + \frac{d}{d h} [- 4 h] + \frac{d}{d h} [h^{2}]) + \frac{d}{d h} [- 27]) - (8 (4 - 4 h + h^{2}) - 27) \frac{d}{d h} [h]}{h^{2}}$

Step 3.4

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

$\frac{h (8 (0 + \frac{d}{d h} [- 4 h] + \frac{d}{d h} [h^{2}]) + \frac{d}{d h} [- 27]) - (8 (4 - 4 h + h^{2}) - 27) \frac{d}{d h} [h]}{h^{2}}$

Step 3.5

Add $0$ and $\frac{d}{d h} [- 4 h]$ .

$\frac{h (8 (\frac{d}{d h} [- 4 h] + \frac{d}{d h} [h^{2}]) + \frac{d}{d h} [- 27]) - (8 (4 - 4 h + h^{2}) - 27) \frac{d}{d h} [h]}{h^{2}}$

Step 3.6

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

$\frac{h (8 (- 4 \frac{d}{d h} [h] + \frac{d}{d h} [h^{2}]) + \frac{d}{d h} [- 27]) - (8 (4 - 4 h + h^{2}) - 27) \frac{d}{d h} [h]}{h^{2}}$

Step 3.7

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

$\frac{h (8 (- 4 \cdot 1 + \frac{d}{d h} [h^{2}]) + \frac{d}{d h} [- 27]) - (8 (4 - 4 h + h^{2}) - 27) \frac{d}{d h} [h]}{h^{2}}$

Step 3.8

Multiply $- 4$ by $1$ .

$\frac{h (8 (- 4 + \frac{d}{d h} [h^{2}]) + \frac{d}{d h} [- 27]) - (8 (4 - 4 h + h^{2}) - 27) \frac{d}{d h} [h]}{h^{2}}$

Step 3.9

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

$\frac{h (8 (- 4 + 2 h) + \frac{d}{d h} [- 27]) - (8 (4 - 4 h + h^{2}) - 27) \frac{d}{d h} [h]}{h^{2}}$

Step 3.10

Since $- 27$ is constant with respect to $h$ , the derivative of $- 27$ with respect to $h$ is $0$ .

$\frac{h (8 (- 4 + 2 h) + 0) - (8 (4 - 4 h + h^{2}) - 27) \frac{d}{d h} [h]}{h^{2}}$

Step 3.11

Add $8 (- 4 + 2 h)$ and $0$ .

$\frac{h (8 (- 4 + 2 h)) - (8 (4 - 4 h + h^{2}) - 27) \frac{d}{d h} [h]}{h^{2}}$

Step 3.12

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

$\frac{h (8 (- 4 + 2 h)) - (8 (4 - 4 h + h^{2}) - 27) \cdot 1}{h^{2}}$

Step 3.13

Multiply $- 1$ by $1$ .

$\frac{h (8 (- 4 + 2 h)) - (8 (4 - 4 h + h^{2}) - 27)}{h^{2}}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\frac{h (8 \cdot - 4 + 8 (2 h)) - (8 (4 - 4 h + h^{2}) - 27)}{h^{2}}$

Step 4.2

Apply the distributive property.

$\frac{h (8 \cdot - 4) + h (8 (2 h)) - (8 (4 - 4 h + h^{2}) - 27)}{h^{2}}$

Step 4.3

Apply the distributive property.

$\frac{h (8 \cdot - 4) + h (8 (2 h)) - (8 \cdot 4 + 8 (- 4 h) + 8 h^{2} - 27)}{h^{2}}$

Step 4.4

Apply the distributive property.

$\frac{h (8 \cdot - 4) + h (8 (2 h)) - (8 \cdot 4) - (8 (- 4 h)) - (8 h^{2}) - - 27}{h^{2}}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Simplify each term.

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Step 4.5.1.1

Multiply $8$ by $- 4$ .

$\frac{h \cdot - 32 + h (8 (2 h)) - (8 \cdot 4) - (8 (- 4 h)) - (8 h^{2}) - - 27}{h^{2}}$

Step 4.5.1.2

Move $- 32$ to the left of $h$ .

$\frac{- 32 \cdot h + h (8 (2 h)) - (8 \cdot 4) - (8 (- 4 h)) - (8 h^{2}) - - 27}{h^{2}}$

Step 4.5.1.3

Rewrite using the commutative property of multiplication.

$\frac{- 32 h + 8 \cdot 2 h \cdot h - (8 \cdot 4) - (8 (- 4 h)) - (8 h^{2}) - - 27}{h^{2}}$

Step 4.5.1.4

Multiply $h$ by $h$ by adding the exponents.

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Step 4.5.1.4.1

Move $h$ .

$\frac{- 32 h + 8 \cdot 2 (h \cdot h) - (8 \cdot 4) - (8 (- 4 h)) - (8 h^{2}) - - 27}{h^{2}}$

Step 4.5.1.4.2

Multiply $h$ by $h$ .

$\frac{- 32 h + 8 \cdot 2 h^{2} - (8 \cdot 4) - (8 (- 4 h)) - (8 h^{2}) - - 27}{h^{2}}$

Step 4.5.1.5

Multiply $8$ by $2$ .

$\frac{- 32 h + 16 h^{2} - (8 \cdot 4) - (8 (- 4 h)) - (8 h^{2}) - - 27}{h^{2}}$

Step 4.5.1.6

Multiply $- (8 \cdot 4)$ .

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Step 4.5.1.6.1

Multiply $8$ by $4$ .

$\frac{- 32 h + 16 h^{2} - 1 \cdot 32 - (8 (- 4 h)) - (8 h^{2}) - - 27}{h^{2}}$

Step 4.5.1.6.2

Multiply $- 1$ by $32$ .

$\frac{- 32 h + 16 h^{2} - 32 - (8 (- 4 h)) - (8 h^{2}) - - 27}{h^{2}}$

Step 4.5.1.7

Multiply $- 4$ by $8$ .

$\frac{- 32 h + 16 h^{2} - 32 - (- 32 h) - (8 h^{2}) - - 27}{h^{2}}$

Step 4.5.1.8

Multiply $- 32$ by $- 1$ .

$\frac{- 32 h + 16 h^{2} - 32 + 32 h - (8 h^{2}) - - 27}{h^{2}}$

Step 4.5.1.9

Multiply $8$ by $- 1$ .

$\frac{- 32 h + 16 h^{2} - 32 + 32 h - 8 h^{2} - - 27}{h^{2}}$

Step 4.5.1.10

Multiply $- 1$ by $- 27$ .

$\frac{- 32 h + 16 h^{2} - 32 + 32 h - 8 h^{2} + 27}{h^{2}}$

Step 4.5.2

Combine the opposite terms in $- 32 h + 16 h^{2} - 32 + 32 h - 8 h^{2} + 27$ .

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Step 4.5.2.1

Add $- 32 h$ and $32 h$ .

$\frac{16 h^{2} - 32 + 0 - 8 h^{2} + 27}{h^{2}}$

Step 4.5.2.2

Add $16 h^{2} - 32$ and $0$ .

$\frac{16 h^{2} - 32 - 8 h^{2} + 27}{h^{2}}$

Step 4.5.3

Subtract $8 h^{2}$ from $16 h^{2}$ .

$\frac{8 h^{2} - 32 + 27}{h^{2}}$

Step 4.5.4

Add $- 32$ and $27$ .

$\frac{8 h^{2} - 5}{h^{2}}$