Calculus Examples

$\frac{(7 {(- 1 + h)}^{2} + 5 (- 1 + h)) - (7 {(- 1)}^{2} + 5 (- 1))}{h}$

Step 1

Simplify the numerator.

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Step 1.1

Let $u = - 1 + h$ . Substitute $u$ for all occurrences of $- 1 + h$ .

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

Raise $- 1$ to the power of $2$ .

$\frac{7 u^{2} + 5 u - (7 \cdot 1 + 5 (- 1))}{h}$

Step 1.1.1.2

Multiply $7$ by $1$ .

$\frac{7 u^{2} + 5 u - (7 + 5 (- 1))}{h}$

Step 1.1.1.3

Multiply $5$ by $- 1$ .

$\frac{7 u^{2} + 5 u - (7 - 5)}{h}$

Step 1.1.2

Subtract $5$ from $7$ .

$\frac{7 u^{2} + 5 u - 1 \cdot 2}{h}$

Step 1.1.3

Multiply $- 1$ by $2$ .

$\frac{7 u^{2} + 5 u - 2}{h}$

Step 1.2

Factor by grouping.

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Step 1.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 7 \cdot - 2 = - 14$ and whose sum is $b = 5$ .

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Step 1.2.1.1

Factor $5$ out of $5 u$ .

$\frac{7 u^{2} + 5 (u) - 2}{h}$

Step 1.2.1.2

Rewrite $5$ as $- 2$ plus $7$

$\frac{7 u^{2} + (- 2 + 7) u - 2}{h}$

Step 1.2.1.3

Apply the distributive property.

$\frac{7 u^{2} - 2 u + 7 u - 2}{h}$

Step 1.2.2

Factor out the greatest common factor from each group.

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Step 1.2.2.1

Group the first two terms and the last two terms.

$\frac{(7 u^{2} - 2 u) + 7 u - 2}{h}$

Step 1.2.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{u (7 u - 2) + 1 (7 u - 2)}{h}$

Step 1.2.3

Factor the polynomial by factoring out the greatest common factor, $7 u - 2$ .

$\frac{(7 u - 2) (u + 1)}{h}$

Step 1.3

Replace all occurrences of $u$ with $- 1 + h$ .

$\frac{(7 (- 1 + h) - 2) (- 1 + h + 1)}{h}$

Step 1.4

Simplify.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Apply the distributive property.

$\frac{(7 \cdot - 1 + 7 h - 2) (- 1 + h + 1)}{h}$

Step 1.4.1.2

Multiply $7$ by $- 1$ .

$\frac{(- 7 + 7 h - 2) (- 1 + h + 1)}{h}$

Step 1.4.2

Subtract $2$ from $- 7$ .

$\frac{(7 h - 9) (- 1 + h + 1)}{h}$

Step 1.4.3

Combine the opposite terms in $- 1 + h + 1$ .

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Step 1.4.3.1

Add $- 1$ and $1$ .

$\frac{(7 h - 9) (h + 0)}{h}$

Step 1.4.3.2

Add $h$ and $0$ .

$\frac{(7 h - 9) h}{h}$

Step 2

Cancel the common factor of $h$ .

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Step 2.1

Cancel the common factor.

$\frac{(7 h - 9) h}{h}$

Step 2.2

Divide $7 h - 9$ by $1$ .

$7 h - 9$