Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

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

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

Step 1.2

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 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.3

Differentiate.

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

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

Step 1.3.5

Multiply.

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Step 1.3.5.1

Multiply $- 1$ by $- 1$ .

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

Step 1.3.5.2

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

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

Step 1.3.6

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

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

Step 1.3.7

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

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

Step 1.4

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

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

Step 1.5

Simplify.

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Step 1.5.1

Combine terms.

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Step 1.5.1.1

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

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

Step 1.5.1.2

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

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

Step 1.5.2

Reorder terms.

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

Step 2

Find the second derivative.

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Step 2.1

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2 x}{{(- x^{2} + 1)}^{2}}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{x}{{(- x^{2} + 1)}^{2}}]$ .

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

Step 2.2

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

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

Step 2.3

Differentiate using the Power Rule.

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Step 2.3.1

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

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Step 2.3.1.1

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

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

Step 2.3.1.2

Multiply $2$ by $2$ .

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

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

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

Step 2.5

Simplify with factoring out.

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Step 2.5.1

Multiply $2$ by $- 1$ .

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

Step 2.5.2

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

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Step 2.5.2.1

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

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

Step 2.5.2.2

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

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

Step 2.5.2.3

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

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

Step 2.6

Cancel the common factors.

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Step 2.6.1

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

$2 \frac{(- x^{2} + 1) (- x^{2} + 1 - 2 x (\frac{d}{d x} [- x^{2} + 1]))}{(- x^{2} + 1) {(- x^{2} + 1)}^{3}}$

Step 2.6.2

Cancel the common factor.

$2 \frac{(- x^{2} + 1) (- x^{2} + 1 - 2 x (\frac{d}{d x} [- x^{2} + 1]))}{(- x^{2} + 1) {(- x^{2} + 1)}^{3}}$

Step 2.6.3

Rewrite the expression.

$2 \frac{- x^{2} + 1 - 2 x (\frac{d}{d x} [- x^{2} + 1])}{{(- x^{2} + 1)}^{3}}$

Step 2.7

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

$2 \frac{- x^{2} + 1 - 2 x (\frac{d}{d x} [- x^{2}] + \frac{d}{d x} [1])}{{(- x^{2} + 1)}^{3}}$

Step 2.8

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$2 \frac{- x^{2} + 1 - 2 x (- \frac{d}{d x} [x^{2}] + \frac{d}{d x} [1])}{{(- x^{2} + 1)}^{3}}$

Step 2.9

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

$2 \frac{- x^{2} + 1 - 2 x (- (2 x) + \frac{d}{d x} [1])}{{(- x^{2} + 1)}^{3}}$

Step 2.10

Multiply $2$ by $- 1$ .

$2 \frac{- x^{2} + 1 - 2 x (- 2 x + \frac{d}{d x} [1])}{{(- x^{2} + 1)}^{3}}$

Step 2.11

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

$2 \frac{- x^{2} + 1 - 2 x (- 2 x + 0)}{{(- x^{2} + 1)}^{3}}$

Step 2.12

Simplify the expression.

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Step 2.12.1

Add $- 2 x$ and $0$ .

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

Step 2.12.2

Multiply $- 2$ by $- 2$ .

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

Step 2.13

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

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

Step 2.14

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

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

Step 2.15

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

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

Step 2.16

Add $1$ and $1$ .

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

Step 2.17

Add $- x^{2}$ and $4 x^{2}$ .

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

Step 2.18

Combine $2$ and $\frac{3 x^{2} + 1}{{(- x^{2} + 1)}^{3}}$ .

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

Step 2.19

Simplify.

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Step 2.19.1

Apply the distributive property.

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

Step 2.19.2

Simplify each term.

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Step 2.19.2.1

Multiply $3$ by $2$ .

$\frac{6 x^{2} + 2 \cdot 1}{{(- x^{2} + 1)}^{3}}$

Step 2.19.2.2

Multiply $2$ by $1$ .

$f'' (x) = \frac{6 x^{2} + 2}{{(- x^{2} + 1)}^{3}}$