Calculus Examples

$\frac{1}{x + 1}$

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Add $1$ and $0$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

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

Multiply $x$ by $0$ .

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

Step 3.2.1.2

Multiply $0$ by $1$ .

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

Step 3.2.1.3

Multiply $- 1 \cdot 1 \cdot 1$ .

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Step 3.2.1.3.1

Multiply $- 1$ by $1$ .

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

Step 3.2.1.3.2

Multiply $- 1$ by $1$ .

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

Step 3.2.2

Add $0$ and $0$ .

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

Step 3.2.3

Subtract $1$ from $0$ .

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

Step 3.3

Move the negative in front of the fraction.

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