Calculus Examples

$\frac{2 x^{2} + x - 3}{2 x + 7}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 2 x^{2} + x - 3$ and $g (x) = 2 x + 7$ .

$\frac{(2 x + 7) \frac{d}{d x} [2 x^{2} + x - 3] - (2 x^{2} + x - 3) \frac{d}{d x} [2 x + 7]}{{(2 x + 7)}^{2}}$

Step 2

Differentiate.

Tap for more steps...

Step 2.1

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

$\frac{(2 x + 7) (\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [- 3]) - (2 x^{2} + x - 3) \frac{d}{d x} [2 x + 7]}{{(2 x + 7)}^{2}}$

Step 2.2

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

$\frac{(2 x + 7) (2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [- 3]) - (2 x^{2} + x - 3) \frac{d}{d x} [2 x + 7]}{{(2 x + 7)}^{2}}$

Step 2.3

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

$\frac{(2 x + 7) (2 (2 x) + \frac{d}{d x} [x] + \frac{d}{d x} [- 3]) - (2 x^{2} + x - 3) \frac{d}{d x} [2 x + 7]}{{(2 x + 7)}^{2}}$

Step 2.4

Multiply $2$ by $2$ .

$\frac{(2 x + 7) (4 x + \frac{d}{d x} [x] + \frac{d}{d x} [- 3]) - (2 x^{2} + x - 3) \frac{d}{d x} [2 x + 7]}{{(2 x + 7)}^{2}}$

Step 2.5

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

$\frac{(2 x + 7) (4 x + 1 + \frac{d}{d x} [- 3]) - (2 x^{2} + x - 3) \frac{d}{d x} [2 x + 7]}{{(2 x + 7)}^{2}}$

Step 2.6

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$\frac{(2 x + 7) (4 x + 1 + 0) - (2 x^{2} + x - 3) \frac{d}{d x} [2 x + 7]}{{(2 x + 7)}^{2}}$

Step 2.7

Add $4 x + 1$ and $0$ .

$\frac{(2 x + 7) (4 x + 1) - (2 x^{2} + x - 3) \frac{d}{d x} [2 x + 7]}{{(2 x + 7)}^{2}}$

Step 2.8

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

$\frac{(2 x + 7) (4 x + 1) - (2 x^{2} + x - 3) (\frac{d}{d x} [2 x] + \frac{d}{d x} [7])}{{(2 x + 7)}^{2}}$

Step 2.9

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

$\frac{(2 x + 7) (4 x + 1) - (2 x^{2} + x - 3) (2 \frac{d}{d x} [x] + \frac{d}{d x} [7])}{{(2 x + 7)}^{2}}$

Step 2.10

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

$\frac{(2 x + 7) (4 x + 1) - (2 x^{2} + x - 3) (2 \cdot 1 + \frac{d}{d x} [7])}{{(2 x + 7)}^{2}}$

Step 2.11

Multiply $2$ by $1$ .

$\frac{(2 x + 7) (4 x + 1) - (2 x^{2} + x - 3) (2 + \frac{d}{d x} [7])}{{(2 x + 7)}^{2}}$

Step 2.12

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$\frac{(2 x + 7) (4 x + 1) - (2 x^{2} + x - 3) (2 + 0)}{{(2 x + 7)}^{2}}$

Step 2.13

Simplify the expression.

Tap for more steps...

Step 2.13.1

Add $2$ and $0$ .

$\frac{(2 x + 7) (4 x + 1) - (2 x^{2} + x - 3) \cdot 2}{{(2 x + 7)}^{2}}$

Step 2.13.2

Multiply $2$ by $- 1$ .

$\frac{(2 x + 7) (4 x + 1) - 2 (2 x^{2} + x - 3)}{{(2 x + 7)}^{2}}$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Apply the distributive property.

$\frac{(2 x + 7) (4 x + 1) - 2 (2 x^{2}) - 2 x - 2 \cdot - 3}{{(2 x + 7)}^{2}}$

Step 3.2

Simplify the numerator.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

Expand $(2 x + 7) (4 x + 1)$ using the FOIL Method.

Tap for more steps...

Step 3.2.1.1.1

Apply the distributive property.

$\frac{2 x (4 x + 1) + 7 (4 x + 1) - 2 (2 x^{2}) - 2 x - 2 \cdot - 3}{{(2 x + 7)}^{2}}$

Step 3.2.1.1.2

Apply the distributive property.

$\frac{2 x (4 x) + 2 x \cdot 1 + 7 (4 x + 1) - 2 (2 x^{2}) - 2 x - 2 \cdot - 3}{{(2 x + 7)}^{2}}$

Step 3.2.1.1.3

Apply the distributive property.

$\frac{2 x (4 x) + 2 x \cdot 1 + 7 (4 x) + 7 \cdot 1 - 2 (2 x^{2}) - 2 x - 2 \cdot - 3}{{(2 x + 7)}^{2}}$

Step 3.2.1.2

Simplify and combine like terms.

Tap for more steps...

Step 3.2.1.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 \cdot 4 x \cdot x + 2 x \cdot 1 + 7 (4 x) + 7 \cdot 1 - 2 (2 x^{2}) - 2 x - 2 \cdot - 3}{{(2 x + 7)}^{2}}$

Step 3.2.1.2.1.2

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 3.2.1.2.1.2.1

Move $x$ .

$\frac{2 \cdot 4 (x \cdot x) + 2 x \cdot 1 + 7 (4 x) + 7 \cdot 1 - 2 (2 x^{2}) - 2 x - 2 \cdot - 3}{{(2 x + 7)}^{2}}$

Step 3.2.1.2.1.2.2

Multiply $x$ by $x$ .

$\frac{2 \cdot 4 x^{2} + 2 x \cdot 1 + 7 (4 x) + 7 \cdot 1 - 2 (2 x^{2}) - 2 x - 2 \cdot - 3}{{(2 x + 7)}^{2}}$

Step 3.2.1.2.1.3

Multiply $2$ by $4$ .

$\frac{8 x^{2} + 2 x \cdot 1 + 7 (4 x) + 7 \cdot 1 - 2 (2 x^{2}) - 2 x - 2 \cdot - 3}{{(2 x + 7)}^{2}}$

Step 3.2.1.2.1.4

Multiply $2$ by $1$ .

$\frac{8 x^{2} + 2 x + 7 (4 x) + 7 \cdot 1 - 2 (2 x^{2}) - 2 x - 2 \cdot - 3}{{(2 x + 7)}^{2}}$

Step 3.2.1.2.1.5

Multiply $4$ by $7$ .

$\frac{8 x^{2} + 2 x + 28 x + 7 \cdot 1 - 2 (2 x^{2}) - 2 x - 2 \cdot - 3}{{(2 x + 7)}^{2}}$

Step 3.2.1.2.1.6

Multiply $7$ by $1$ .

$\frac{8 x^{2} + 2 x + 28 x + 7 - 2 (2 x^{2}) - 2 x - 2 \cdot - 3}{{(2 x + 7)}^{2}}$

Step 3.2.1.2.2

Add $2 x$ and $28 x$ .

$\frac{8 x^{2} + 30 x + 7 - 2 (2 x^{2}) - 2 x - 2 \cdot - 3}{{(2 x + 7)}^{2}}$

Step 3.2.1.3

Multiply $2$ by $- 2$ .

$\frac{8 x^{2} + 30 x + 7 - 4 x^{2} - 2 x - 2 \cdot - 3}{{(2 x + 7)}^{2}}$

Step 3.2.1.4

Multiply $- 2$ by $- 3$ .

$\frac{8 x^{2} + 30 x + 7 - 4 x^{2} - 2 x + 6}{{(2 x + 7)}^{2}}$

Step 3.2.2

Subtract $4 x^{2}$ from $8 x^{2}$ .

$\frac{4 x^{2} + 30 x + 7 - 2 x + 6}{{(2 x + 7)}^{2}}$

Step 3.2.3

Subtract $2 x$ from $30 x$ .

$\frac{4 x^{2} + 28 x + 7 + 6}{{(2 x + 7)}^{2}}$

Step 3.2.4

Add $7$ and $6$ .

$\frac{4 x^{2} + 28 x + 13}{{(2 x + 7)}^{2}}$

Step 3.3

Factor by grouping.

Tap for more steps...

Step 3.3.1

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 $a \cdot c = 4 \cdot 13 = 52$ and whose sum is $b = 28$ .

Tap for more steps...

Step 3.3.1.1

Factor $28$ out of $28 x$ .

$\frac{4 x^{2} + 28 (x) + 13}{{(2 x + 7)}^{2}}$

Step 3.3.1.2

Rewrite $28$ as $2$ plus $26$

$\frac{4 x^{2} + (2 + 26) x + 13}{{(2 x + 7)}^{2}}$

Step 3.3.1.3

Apply the distributive property.

$\frac{4 x^{2} + 2 x + 26 x + 13}{{(2 x + 7)}^{2}}$

Step 3.3.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 3.3.2.1

Group the first two terms and the last two terms.

$\frac{(4 x^{2} + 2 x) + 26 x + 13}{{(2 x + 7)}^{2}}$

Step 3.3.2.2

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

$\frac{2 x (2 x + 1) + 13 (2 x + 1)}{{(2 x + 7)}^{2}}$

Step 3.3.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

$\frac{(2 x + 1) (2 x + 13)}{{(2 x + 7)}^{2}}$