Calculus Examples

$\frac{x^{2} + m x + 1}{x + m}$

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 $f (x) = x^{2} + m x + 1$ and $g (x) = x + m$ .

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

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x^{2} + m x + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [m x] + \frac{d}{d x} [1]$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Multiply $m$ by $1$ .

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

Step 2.6

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

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

Step 2.7

Add $2 x + m$ and $0$ .

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

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

$\frac{(x + m) (2 x + m) - (x^{2} + m x + 1) (1 + 0)}{{(x + m)}^{2}}$

Step 2.11

Simplify the expression.

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Step 2.11.1

Add $1$ and $0$ .

$\frac{(x + m) (2 x + m) - (x^{2} + m x + 1) \cdot 1}{{(x + m)}^{2}}$

Step 2.11.2

Multiply $- 1$ by $1$ .

$\frac{(x + m) (2 x + m) - (x^{2} + m x + 1)}{{(x + m)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{(x + m) (2 x + m) - x^{2} - (m x) - 1 \cdot 1}{{(x + m)}^{2}}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Expand $(x + m) (2 x + m)$ using the FOIL Method.

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Step 3.2.1.1.1

Apply the distributive property.

$\frac{x (2 x + m) + m (2 x + m) - x^{2} - (m x) - 1 \cdot 1}{{(x + m)}^{2}}$

Step 3.2.1.1.2

Apply the distributive property.

$\frac{x (2 x) + x m + m (2 x + m) - x^{2} - (m x) - 1 \cdot 1}{{(x + m)}^{2}}$

Step 3.2.1.1.3

Apply the distributive property.

$\frac{x (2 x) + x m + m (2 x) + m \cdot m - x^{2} - (m x) - 1 \cdot 1}{{(x + m)}^{2}}$

Step 3.2.1.2

Simplify and combine like terms.

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Step 3.2.1.2.1

Simplify each term.

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Step 3.2.1.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 x \cdot x + x m + m (2 x) + m \cdot m - x^{2} - (m x) - 1 \cdot 1}{{(x + m)}^{2}}$

Step 3.2.1.2.1.2

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

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Step 3.2.1.2.1.2.1

Move $x$ .

$\frac{2 (x \cdot x) + x m + m (2 x) + m \cdot m - x^{2} - (m x) - 1 \cdot 1}{{(x + m)}^{2}}$

Step 3.2.1.2.1.2.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} + x m + m (2 x) + m \cdot m - x^{2} - (m x) - 1 \cdot 1}{{(x + m)}^{2}}$

Step 3.2.1.2.1.3

Rewrite using the commutative property of multiplication.

$\frac{2 x^{2} + x m + 2 m x + m \cdot m - x^{2} - (m x) - 1 \cdot 1}{{(x + m)}^{2}}$

Step 3.2.1.2.1.4

Multiply $m$ by $m$ .

$\frac{2 x^{2} + x m + 2 m x + m^{2} - x^{2} - (m x) - 1 \cdot 1}{{(x + m)}^{2}}$

Step 3.2.1.2.2

Add $x m$ and $2 m x$ .

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Step 3.2.1.2.2.1

Reorder $x$ and $m$ .

$\frac{2 x^{2} + m x + 2 m x + m^{2} - x^{2} - (m x) - 1 \cdot 1}{{(x + m)}^{2}}$

Step 3.2.1.2.2.2

Add $m x$ and $2 m x$ .

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

Step 3.2.1.3

Multiply $- 1$ by $1$ .

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

Step 3.2.2

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

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

Step 3.2.3

Subtract $m x$ from $3 m x$ .

$\frac{x^{2} + m^{2} + 2 m x - 1}{{(x + m)}^{2}}$

Step 3.3

Reorder terms.

$\frac{m^{2} + x^{2} + 2 m x - 1}{{(m + x)}^{2}}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Factor using the perfect square rule.

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Step 3.4.1.1

Rearrange terms.

$\frac{m^{2} + 2 m x + x^{2} - 1}{{(m + x)}^{2}}$

Step 3.4.1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 m x = 2 \cdot m \cdot x$

Step 3.4.1.3

Rewrite the polynomial.

$\frac{m^{2} + 2 \cdot m \cdot x + x^{2} - 1}{{(m + x)}^{2}}$

Step 3.4.1.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = m$ and $b = x$ .

$\frac{{(m + x)}^{2} - 1}{{(m + x)}^{2}}$

Step 3.4.2

Rewrite $1$ as $1^{2}$ .

$\frac{{(m + x)}^{2} - 1^{2}}{{(m + x)}^{2}}$

Step 3.4.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = m + x$ and $b = 1$ .

$\frac{(m + x + 1) (m + x - 1)}{{(m + x)}^{2}}$