Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

Differentiate.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

Simplify the expression.

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Step 3.2.4.1

Add $1$ and $0$ .

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

Step 3.2.4.2

Multiply $x + 2$ by $1$ .

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

Step 3.2.5

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

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

Step 3.2.6

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

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

Step 3.2.7

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

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

Step 3.2.8

Simplify the expression.

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Step 3.2.8.1

Add $1$ and $0$ .

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

Step 3.2.8.2

Multiply $- 1$ by $1$ .

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

Step 3.3

Simplify.

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Step 3.3.1

Apply the distributive property.

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

Step 3.3.2

Simplify the numerator.

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Step 3.3.2.1

Combine the opposite terms in $x + 2 - x - 1 \cdot 1$ .

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Step 3.3.2.1.1

Subtract $x$ from $x$ .

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

Step 3.3.2.1.2

Add $0$ and $2$ .

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

Step 3.3.2.2

Multiply $- 1$ by $1$ .

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

Step 3.3.2.3

Subtract $1$ from $2$ .

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

Step 4

Reform the equation by setting the left side equal to the right side.

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

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

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