Calculus Examples

$y = (x + 6) (\frac{x}{x + 1})$

Step 1

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 $f (x) = x + 6$ and $g (x) = \frac{x}{x + 1}$ .

$(x + 6) \frac{d}{d x} [\frac{x}{x + 1}] + \frac{x}{x + 1} \frac{d}{d x} [x + 6]$

Step 2

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

$(x + 6) \frac{(x + 1) \frac{d}{d x} [x] - x \frac{d}{d x} [x + 1]}{{(x + 1)}^{2}} + \frac{x}{x + 1} \frac{d}{d x} [x + 6]$

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 $n = 1$ .

$(x + 6) \frac{(x + 1) \cdot 1 - x \frac{d}{d x} [x + 1]}{{(x + 1)}^{2}} + \frac{x}{x + 1} \frac{d}{d x} [x + 6]$

Step 3.2

Multiply $x + 1$ by $1$ .

$(x + 6) \frac{x + 1 - x \frac{d}{d x} [x + 1]}{{(x + 1)}^{2}} + \frac{x}{x + 1} \frac{d}{d x} [x + 6]$

Step 3.3

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

$(x + 6) \frac{x + 1 - x (\frac{d}{d x} [x] + \frac{d}{d x} [1])}{{(x + 1)}^{2}} + \frac{x}{x + 1} \frac{d}{d x} [x + 6]$

Step 3.4

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

$(x + 6) \frac{x + 1 - x (1 + \frac{d}{d x} [1])}{{(x + 1)}^{2}} + \frac{x}{x + 1} \frac{d}{d x} [x + 6]$

Step 3.5

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

$(x + 6) \frac{x + 1 - x (1 + 0)}{{(x + 1)}^{2}} + \frac{x}{x + 1} \frac{d}{d x} [x + 6]$

Step 3.6

Simplify by adding terms.

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Step 3.6.1

Add $1$ and $0$ .

$(x + 6) \frac{x + 1 - x \cdot 1}{{(x + 1)}^{2}} + \frac{x}{x + 1} \frac{d}{d x} [x + 6]$

Step 3.6.2

Multiply $- 1$ by $1$ .

$(x + 6) \frac{x + 1 - x}{{(x + 1)}^{2}} + \frac{x}{x + 1} \frac{d}{d x} [x + 6]$

Step 3.6.3

Subtract $x$ from $x$ .

$(x + 6) \frac{0 + 1}{{(x + 1)}^{2}} + \frac{x}{x + 1} \frac{d}{d x} [x + 6]$

Step 3.6.4

Add $0$ and $1$ .

$(x + 6) \frac{1}{{(x + 1)}^{2}} + \frac{x}{x + 1} \frac{d}{d x} [x + 6]$

Step 3.7

By the Sum Rule, the derivative of $x + 6$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [6]$ .

$(x + 6) \frac{1}{{(x + 1)}^{2}} + \frac{x}{x + 1} (\frac{d}{d x} [x] + \frac{d}{d x} [6])$

Step 3.8

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

$(x + 6) \frac{1}{{(x + 1)}^{2}} + \frac{x}{x + 1} (1 + \frac{d}{d x} [6])$

Step 3.9

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$(x + 6) \frac{1}{{(x + 1)}^{2}} + \frac{x}{x + 1} (1 + 0)$

Step 3.10

Simplify the expression.

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Step 3.10.1

Add $1$ and $0$ .

$(x + 6) \frac{1}{{(x + 1)}^{2}} + \frac{x}{x + 1} \cdot 1$

Step 3.10.2

Multiply $\frac{x}{x + 1}$ by $1$ .

$(x + 6) \frac{1}{{(x + 1)}^{2}} + \frac{x}{x + 1}$

Step 4

To write $(x + 6) \frac{1}{{(x + 1)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

$(x + 6) \frac{1}{{(x + 1)}^{2}} \cdot \frac{x + 1}{x + 1} + \frac{x}{x + 1}$

Step 5

Combine $(x + 6) \frac{1}{{(x + 1)}^{2}}$ and $\frac{x + 1}{x + 1}$ .

$\frac{(x + 6) \frac{1}{{(x + 1)}^{2}} (x + 1)}{x + 1} + \frac{x}{x + 1}$

Step 6

Combine the numerators over the common denominator.

$\frac{(x + 6) \frac{1}{{(x + 1)}^{2}} (x + 1) + x}{x + 1}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Simplify each term.

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Step 7.1.1.1

Multiply $x + 6$ by $\frac{1}{{(x + 1)}^{2}}$ .

$\frac{\frac{x + 6}{{(x + 1)}^{2}} (x + 1) + x}{x + 1}$

Step 7.1.1.2

Cancel the common factor of $x + 1$ .

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Step 7.1.1.2.1

Factor $x + 1$ out of ${(x + 1)}^{2}$ .

$\frac{\frac{x + 6}{(x + 1) (x + 1)} (x + 1) + x}{x + 1}$

Step 7.1.1.2.2

Cancel the common factor.

$\frac{\frac{x + 6}{(x + 1) (x + 1)} (x + 1) + x}{x + 1}$

Step 7.1.1.2.3

Rewrite the expression.

$\frac{\frac{x + 6}{x + 1} + x}{x + 1}$

Step 7.1.2

To write $x$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

$\frac{\frac{x + 6}{x + 1} + \frac{x (x + 1)}{x + 1}}{x + 1}$

Step 7.1.3

Combine the numerators over the common denominator.

$\frac{\frac{x + 6 + x (x + 1)}{x + 1}}{x + 1}$

Step 7.1.4

Simplify the numerator.

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Step 7.1.4.1

Apply the distributive property.

$\frac{\frac{x + 6 + x \cdot x + x \cdot 1}{x + 1}}{x + 1}$

Step 7.1.4.2

Multiply $x$ by $x$ .

$\frac{\frac{x + 6 + x^{2} + x \cdot 1}{x + 1}}{x + 1}$

Step 7.1.4.3

Multiply $x$ by $1$ .

$\frac{\frac{x + 6 + x^{2} + x}{x + 1}}{x + 1}$

Step 7.1.4.4

Add $x$ and $x$ .

$\frac{\frac{2 x + 6 + x^{2}}{x + 1}}{x + 1}$

Step 7.1.4.5

Reorder terms.

$\frac{\frac{x^{2} + 2 x + 6}{x + 1}}{x + 1}$

Step 7.2

Combine terms.

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Step 7.2.1

Rewrite $\frac{\frac{x^{2} + 2 x + 6}{x + 1}}{x + 1}$ as a product.

$\frac{x^{2} + 2 x + 6}{x + 1} \cdot \frac{1}{x + 1}$

Step 7.2.2

Multiply $\frac{x^{2} + 2 x + 6}{x + 1}$ by $\frac{1}{x + 1}$ .

$\frac{x^{2} + 2 x + 6}{(x + 1) (x + 1)}$

Step 7.2.3

Raise $x + 1$ to the power of $1$ .

$\frac{x^{2} + 2 x + 6}{{(x + 1)}^{1} (x + 1)}$

Step 7.2.4

Raise $x + 1$ to the power of $1$ .

$\frac{x^{2} + 2 x + 6}{{(x + 1)}^{1} {(x + 1)}^{1}}$

Step 7.2.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{2} + 2 x + 6}{{(x + 1)}^{1 + 1}}$

Step 7.2.6

Add $1$ and $1$ .

$\frac{x^{2} + 2 x + 6}{{(x + 1)}^{2}}$