Calculus Examples

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

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Multiply $2$ by $1$ .

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Add $1$ and $0$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

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

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

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

Step 3.2.1.2

Multiply $2$ by $1$ .

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

Step 3.2.1.3

Multiply $- 1$ by $2$ .

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

Step 3.2.1.4

Multiply $- 2$ by $1$ .

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

Step 3.2.2

Combine the opposite terms in $2 x + 2 - 2 x$ .

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Step 3.2.2.1

Subtract $2 x$ from $2 x$ .

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

Step 3.2.2.2

Add $0$ and $2$ .

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