Calculus Examples

$y = \frac{(x + 2) x}{x + 1}$

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 $f (x) = (x + 2) x$ and $g (x) = x + 1$ .

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

Step 2

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

Multiply $x + 2$ by $1$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

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

Step 3.6

Simplify by adding terms.

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Step 3.6.1

Add $1$ and $0$ .

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

Step 3.6.2

Multiply $x$ by $1$ .

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

Step 3.6.3

Add $x$ and $x$ .

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

Simplify the expression.

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Step 3.10.1

Add $1$ and $0$ .

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

Step 3.10.2

Multiply $- 1$ by $1$ .

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Simplify the numerator.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

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

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Step 4.3.1.1.1

Apply the distributive property.

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

Step 4.3.1.1.2

Apply the distributive property.

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

Step 4.3.1.1.3

Apply the distributive property.

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

Step 4.3.1.2

Simplify and combine like terms.

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Step 4.3.1.2.1

Simplify each term.

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Step 4.3.1.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 4.3.1.2.1.2

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

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Step 4.3.1.2.1.2.1

Move $x$ .

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

Step 4.3.1.2.1.2.2

Multiply $x$ by $x$ .

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

Step 4.3.1.2.1.3

Move $2$ to the left of $x$ .

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

Step 4.3.1.2.1.4

Multiply $2 x$ by $1$ .

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

Step 4.3.1.2.1.5

Multiply $2$ by $1$ .

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

Step 4.3.1.2.2

Add $2 x$ and $2 x$ .

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

Step 4.3.1.3

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

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Step 4.3.1.3.1

Move $x$ .

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

Step 4.3.1.3.2

Multiply $x$ by $x$ .

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

Step 4.3.1.4

Multiply $- 1$ by $2$ .

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

Step 4.3.2

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

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

Step 4.3.3

Subtract $2 x$ from $4 x$ .

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