Calculus Examples

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

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

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

Step 2

Differentiate.

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Step 2.1

Multiply the exponents in ${({(x + 1)}^{2})}^{2}$ .

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Step 2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.1.2

Multiply $2$ by $2$ .

$\frac{{(x + 1)}^{2} \frac{d}{d x} [x^{2} - 1] - (x^{2} - 1) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 2.2

By the Sum Rule, the derivative of $x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]$ .

$\frac{{(x + 1)}^{2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]) - (x^{2} - 1) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 2.3

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

$\frac{{(x + 1)}^{2} (2 x + \frac{d}{d x} [- 1]) - (x^{2} - 1) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 2.4

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

$\frac{{(x + 1)}^{2} (2 x + 0) - (x^{2} - 1) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 2.5

Simplify the expression.

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Step 2.5.1

Add $2 x$ and $0$ .

$\frac{{(x + 1)}^{2} (2 x) - (x^{2} - 1) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 2.5.2

Move $2$ to the left of ${(x + 1)}^{2}$ .

$\frac{2 \cdot {(x + 1)}^{2} x - (x^{2} - 1) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x + 1$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $x + 1$ .

$\frac{2 {(x + 1)}^{2} x - (x^{2} - 1) (\frac{d}{d u} [u^{2}] \frac{d}{d x} [x + 1])}{{(x + 1)}^{4}}$

Step 3.2

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

$\frac{2 {(x + 1)}^{2} x - (x^{2} - 1) (2 u \frac{d}{d x} [x + 1])}{{(x + 1)}^{4}}$

Step 3.3

Replace all occurrences of $u$ with $x + 1$ .

$\frac{2 {(x + 1)}^{2} x - (x^{2} - 1) (2 (x + 1) \frac{d}{d x} [x + 1])}{{(x + 1)}^{4}}$

Step 4

Differentiate.

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Step 4.1

Multiply $2$ by $- 1$ .

$\frac{2 {(x + 1)}^{2} x - 2 (x^{2} - 1) ((x + 1) \frac{d}{d x} [x + 1])}{{(x + 1)}^{4}}$

Step 4.2

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

$\frac{2 {(x + 1)}^{2} x - 2 (x^{2} - 1) ((x + 1) (\frac{d}{d x} [x] + \frac{d}{d x} [1]))}{{(x + 1)}^{4}}$

Step 4.3

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

$\frac{2 {(x + 1)}^{2} x - 2 (x^{2} - 1) ((x + 1) (1 + \frac{d}{d x} [1]))}{{(x + 1)}^{4}}$

Step 4.4

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

$\frac{2 {(x + 1)}^{2} x - 2 (x^{2} - 1) ((x + 1) (1 + 0))}{{(x + 1)}^{4}}$

Step 4.5

Simplify the expression.

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Step 4.5.1

Add $1$ and $0$ .

$\frac{2 {(x + 1)}^{2} x - 2 (x^{2} - 1) ((x + 1) \cdot 1)}{{(x + 1)}^{4}}$

Step 4.5.2

Multiply $x + 1$ by $1$ .

$\frac{2 {(x + 1)}^{2} x - 2 (x^{2} - 1) (x + 1)}{{(x + 1)}^{4}}$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$\frac{2 {(x + 1)}^{2} x + (- 2 x^{2} - 2 \cdot - 1) (x + 1)}{{(x + 1)}^{4}}$

Step 5.2

Simplify the numerator.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

$\frac{2 ((x + 1) (x + 1)) x + (- 2 x^{2} - 2 \cdot - 1) (x + 1)}{{(x + 1)}^{4}}$

Step 5.2.1.2

Expand $(x + 1) (x + 1)$ using the FOIL Method.

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Step 5.2.1.2.1

Apply the distributive property.

$\frac{2 (x (x + 1) + 1 (x + 1)) x + (- 2 x^{2} - 2 \cdot - 1) (x + 1)}{{(x + 1)}^{4}}$

Step 5.2.1.2.2

Apply the distributive property.

$\frac{2 (x \cdot x + x \cdot 1 + 1 (x + 1)) x + (- 2 x^{2} - 2 \cdot - 1) (x + 1)}{{(x + 1)}^{4}}$

Step 5.2.1.2.3

Apply the distributive property.

$\frac{2 (x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1) x + (- 2 x^{2} - 2 \cdot - 1) (x + 1)}{{(x + 1)}^{4}}$

Step 5.2.1.3

Simplify and combine like terms.

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Step 5.2.1.3.1

Simplify each term.

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Step 5.2.1.3.1.1

Multiply $x$ by $x$ .

$\frac{2 (x^{2} + x \cdot 1 + 1 x + 1 \cdot 1) x + (- 2 x^{2} - 2 \cdot - 1) (x + 1)}{{(x + 1)}^{4}}$

Step 5.2.1.3.1.2

Multiply $x$ by $1$ .

$\frac{2 (x^{2} + x + 1 x + 1 \cdot 1) x + (- 2 x^{2} - 2 \cdot - 1) (x + 1)}{{(x + 1)}^{4}}$

Step 5.2.1.3.1.3

Multiply $x$ by $1$ .

$\frac{2 (x^{2} + x + x + 1 \cdot 1) x + (- 2 x^{2} - 2 \cdot - 1) (x + 1)}{{(x + 1)}^{4}}$

Step 5.2.1.3.1.4

Multiply $1$ by $1$ .

$\frac{2 (x^{2} + x + x + 1) x + (- 2 x^{2} - 2 \cdot - 1) (x + 1)}{{(x + 1)}^{4}}$

Step 5.2.1.3.2

Add $x$ and $x$ .

$\frac{2 (x^{2} + 2 x + 1) x + (- 2 x^{2} - 2 \cdot - 1) (x + 1)}{{(x + 1)}^{4}}$

Step 5.2.1.4

Apply the distributive property.

$\frac{(2 x^{2} + 2 (2 x) + 2 \cdot 1) x + (- 2 x^{2} - 2 \cdot - 1) (x + 1)}{{(x + 1)}^{4}}$

Step 5.2.1.5

Simplify.

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Step 5.2.1.5.1

Multiply $2$ by $2$ .

$\frac{(2 x^{2} + 4 x + 2 \cdot 1) x + (- 2 x^{2} - 2 \cdot - 1) (x + 1)}{{(x + 1)}^{4}}$

Step 5.2.1.5.2

Multiply $2$ by $1$ .

$\frac{(2 x^{2} + 4 x + 2) x + (- 2 x^{2} - 2 \cdot - 1) (x + 1)}{{(x + 1)}^{4}}$

Step 5.2.1.6

Apply the distributive property.

$\frac{2 x^{2} x + 4 x \cdot x + 2 x + (- 2 x^{2} - 2 \cdot - 1) (x + 1)}{{(x + 1)}^{4}}$

Step 5.2.1.7

Simplify.

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Step 5.2.1.7.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 5.2.1.7.1.1

Move $x$ .

$\frac{2 (x \cdot x^{2}) + 4 x \cdot x + 2 x + (- 2 x^{2} - 2 \cdot - 1) (x + 1)}{{(x + 1)}^{4}}$

Step 5.2.1.7.1.2

Multiply $x$ by $x^{2}$ .

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Step 5.2.1.7.1.2.1

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

$\frac{2 (x^{1} x^{2}) + 4 x \cdot x + 2 x + (- 2 x^{2} - 2 \cdot - 1) (x + 1)}{{(x + 1)}^{4}}$

Step 5.2.1.7.1.2.2

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

$\frac{2 x^{1 + 2} + 4 x \cdot x + 2 x + (- 2 x^{2} - 2 \cdot - 1) (x + 1)}{{(x + 1)}^{4}}$

Step 5.2.1.7.1.3

Add $1$ and $2$ .

$\frac{2 x^{3} + 4 x \cdot x + 2 x + (- 2 x^{2} - 2 \cdot - 1) (x + 1)}{{(x + 1)}^{4}}$

Step 5.2.1.7.2

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

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Step 5.2.1.7.2.1

Move $x$ .

$\frac{2 x^{3} + 4 (x \cdot x) + 2 x + (- 2 x^{2} - 2 \cdot - 1) (x + 1)}{{(x + 1)}^{4}}$

Step 5.2.1.7.2.2

Multiply $x$ by $x$ .

$\frac{2 x^{3} + 4 x^{2} + 2 x + (- 2 x^{2} - 2 \cdot - 1) (x + 1)}{{(x + 1)}^{4}}$

Step 5.2.1.8

Multiply $- 2$ by $- 1$ .

$\frac{2 x^{3} + 4 x^{2} + 2 x + (- 2 x^{2} + 2) (x + 1)}{{(x + 1)}^{4}}$

Step 5.2.1.9

Expand $(- 2 x^{2} + 2) (x + 1)$ using the FOIL Method.

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Step 5.2.1.9.1

Apply the distributive property.

$\frac{2 x^{3} + 4 x^{2} + 2 x - 2 x^{2} (x + 1) + 2 (x + 1)}{{(x + 1)}^{4}}$

Step 5.2.1.9.2

Apply the distributive property.

$\frac{2 x^{3} + 4 x^{2} + 2 x - 2 x^{2} x - 2 x^{2} \cdot 1 + 2 (x + 1)}{{(x + 1)}^{4}}$

Step 5.2.1.9.3

Apply the distributive property.

$\frac{2 x^{3} + 4 x^{2} + 2 x - 2 x^{2} x - 2 x^{2} \cdot 1 + 2 x + 2 \cdot 1}{{(x + 1)}^{4}}$

Step 5.2.1.10

Simplify each term.

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Step 5.2.1.10.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 5.2.1.10.1.1

Move $x$ .

$\frac{2 x^{3} + 4 x^{2} + 2 x - 2 (x \cdot x^{2}) - 2 x^{2} \cdot 1 + 2 x + 2 \cdot 1}{{(x + 1)}^{4}}$

Step 5.2.1.10.1.2

Multiply $x$ by $x^{2}$ .

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Step 5.2.1.10.1.2.1

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

$\frac{2 x^{3} + 4 x^{2} + 2 x - 2 (x^{1} x^{2}) - 2 x^{2} \cdot 1 + 2 x + 2 \cdot 1}{{(x + 1)}^{4}}$

Step 5.2.1.10.1.2.2

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

$\frac{2 x^{3} + 4 x^{2} + 2 x - 2 x^{1 + 2} - 2 x^{2} \cdot 1 + 2 x + 2 \cdot 1}{{(x + 1)}^{4}}$

Step 5.2.1.10.1.3

Add $1$ and $2$ .

$\frac{2 x^{3} + 4 x^{2} + 2 x - 2 x^{3} - 2 x^{2} \cdot 1 + 2 x + 2 \cdot 1}{{(x + 1)}^{4}}$

Step 5.2.1.10.2

Multiply $- 2$ by $1$ .

$\frac{2 x^{3} + 4 x^{2} + 2 x - 2 x^{3} - 2 x^{2} + 2 x + 2 \cdot 1}{{(x + 1)}^{4}}$

Step 5.2.1.10.3

Multiply $2$ by $1$ .

$\frac{2 x^{3} + 4 x^{2} + 2 x - 2 x^{3} - 2 x^{2} + 2 x + 2}{{(x + 1)}^{4}}$

Step 5.2.2

Combine the opposite terms in $2 x^{3} + 4 x^{2} + 2 x - 2 x^{3} - 2 x^{2} + 2 x + 2$ .

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Step 5.2.2.1

Subtract $2 x^{3}$ from $2 x^{3}$ .

$\frac{4 x^{2} + 2 x + 0 - 2 x^{2} + 2 x + 2}{{(x + 1)}^{4}}$

Step 5.2.2.2

Add $4 x^{2} + 2 x$ and $0$ .

$\frac{4 x^{2} + 2 x - 2 x^{2} + 2 x + 2}{{(x + 1)}^{4}}$

Step 5.2.3

Subtract $2 x^{2}$ from $4 x^{2}$ .

$\frac{2 x^{2} + 2 x + 2 x + 2}{{(x + 1)}^{4}}$

Step 5.2.4

Add $2 x$ and $2 x$ .

$\frac{2 x^{2} + 4 x + 2}{{(x + 1)}^{4}}$

Step 5.3

Simplify the numerator.

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Step 5.3.1

Factor $2$ out of $2 x^{2} + 4 x + 2$ .

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Step 5.3.1.1

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

$\frac{2 (x^{2}) + 4 x + 2}{{(x + 1)}^{4}}$

Step 5.3.1.2

Factor $2$ out of $4 x$ .

$\frac{2 (x^{2}) + 2 (2 x) + 2}{{(x + 1)}^{4}}$

Step 5.3.1.3

Factor $2$ out of $2$ .

$\frac{2 (x^{2}) + 2 (2 x) + 2 (1)}{{(x + 1)}^{4}}$

Step 5.3.1.4

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

$\frac{2 (x^{2} + 2 x) + 2 (1)}{{(x + 1)}^{4}}$

Step 5.3.1.5

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

$\frac{2 (x^{2} + 2 x + 1)}{{(x + 1)}^{4}}$

Step 5.3.2

Factor using the perfect square rule.

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Step 5.3.2.1

Rewrite $1$ as $1^{2}$ .

$\frac{2 (x^{2} + 2 x + 1^{2})}{{(x + 1)}^{4}}$

Step 5.3.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 5.3.2.3

Rewrite the polynomial.

$\frac{2 (x^{2} + 2 \cdot x \cdot 1 + 1^{2})}{{(x + 1)}^{4}}$

Step 5.3.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 1$ .

$\frac{2 {(x + 1)}^{2}}{{(x + 1)}^{4}}$

Step 5.4

Cancel the common factor of ${(x + 1)}^{2}$ and ${(x + 1)}^{4}$ .

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Step 5.4.1

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

$\frac{{(x + 1)}^{2} \cdot 2}{{(x + 1)}^{4}}$

Step 5.4.2

Cancel the common factors.

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Step 5.4.2.1

Factor ${(x + 1)}^{2}$ out of ${(x + 1)}^{4}$ .

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

Step 5.4.2.2

Cancel the common factor.

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

Step 5.4.2.3

Rewrite the expression.

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