Calculus Examples

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

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

Move $2$ to the left of $x + 1$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $1$ and $0$ .

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

Step 2.6.2

Multiply $- 1$ by $1$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Simplify the numerator.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

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

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Step 3.3.1.1.1

Move $x$ .

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

Step 3.3.1.1.2

Multiply $x$ by $x$ .

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

Step 3.3.1.2

Multiply $2$ by $1$ .

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

Step 3.3.2

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

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

Step 3.4

Factor $x$ out of $x^{2} + 2 x$ .

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Step 3.4.1

Factor $x$ out of $x^{2}$ .

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

Step 3.4.2

Factor $x$ out of $2 x$ .

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

Step 3.4.3

Factor $x$ out of $x \cdot x + x \cdot 2$ .

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