Calculus Examples

$y = (1 - x) {(1 + x^{2})}^{- 1}$

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

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

Step 2

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

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Step 2.1

To apply the Chain Rule, set $u$ as $1 + x^{2}$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $1 + x^{2}$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Add $0$ and $\frac{d}{d x} [x^{2}]$ .

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

Step 3.4

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

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

Step 3.5

Multiply $2$ by $- 1$ .

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

Add $0$ and $\frac{d}{d x} [- x]$ .

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

Step 3.9

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

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

Step 3.10

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

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

Step 3.11

Simplify the expression.

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Step 3.11.1

Multiply $- 1$ by $1$ .

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

Step 3.11.2

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

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

Step 3.11.3

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

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

Step 4

Simplify.

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Step 4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 4.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 4.3

Combine terms.

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Step 4.3.1

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

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

Step 4.3.2

Move the negative in front of the fraction.

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

Step 4.3.3

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

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

Step 4.3.4

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

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

Step 4.3.5

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

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

Step 4.3.6

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

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

Step 4.3.7

Combine the numerators over the common denominator.

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

Step 4.4

Reorder terms.

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

Step 4.5

Simplify the numerator.

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Step 4.5.1

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

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Step 4.5.1.1

Reorder the expression.

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Step 4.5.1.1.1

Reorder $1$ and $x^{2}$ .

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

Step 4.5.1.1.2

Reorder $1$ and $- x$ .

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

Step 4.5.1.1.3

Reorder $1$ and $x^{2}$ .

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

Step 4.5.1.2

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

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

Step 4.5.1.3

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 4.5.1.4

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

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

Step 4.5.2

Cancel the common factor of $x^{2} + 1$ .

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Step 4.5.2.1

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

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

Step 4.5.2.2

Cancel the common factor.

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

Step 4.5.2.3

Rewrite the expression.

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

Step 4.5.3

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

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

Step 4.5.4

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

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

Step 4.5.5

Write $1$ as a fraction with a common denominator.

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

Step 4.5.6

Combine the numerators over the common denominator.

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

Step 4.5.7

Reorder terms.

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

Step 4.5.8

Rewrite $\frac{x^{2} + 2 x (- x + 1) + 1}{x^{2} + 1}$ in a factored form.

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Step 4.5.8.1

Apply the distributive property.

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

Step 4.5.8.2

Rewrite using the commutative property of multiplication.

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

Step 4.5.8.3

Multiply $2$ by $1$ .

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

Step 4.5.8.4

Simplify each term.

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Step 4.5.8.4.1

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

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Step 4.5.8.4.1.1

Move $x$ .

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

Step 4.5.8.4.1.2

Multiply $x$ by $x$ .

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

Step 4.5.8.4.2

Multiply $2$ by $- 1$ .

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

Step 4.5.8.5

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

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

Step 4.6

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.7

Multiply $- \frac{- x^{2} + 2 x + 1}{x^{2} + 1} \cdot \frac{1}{x^{2} + 1}$ .

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Step 4.7.1

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

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

Step 4.7.2

Raise $x^{2} + 1$ to the power of $1$ .

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

Step 4.7.3

Raise $x^{2} + 1$ to the power of $1$ .

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

Step 4.7.4

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

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

Step 4.7.5

Add $1$ and $1$ .

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

Step 4.8

Factor $- 1$ out of $- x^{2}$ .

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

Step 4.9

Factor $- 1$ out of $2 x$ .

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

Step 4.10

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

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

Step 4.11

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 4.12

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

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

Step 4.13

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

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

Step 4.14

Move the negative in front of the fraction.

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

Step 4.15

Multiply $- 1$ by $- 1$ .

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

Step 4.16

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

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