Calculus Examples

$\arcsin (x)$

Step 1

The derivative of $\arcsin (x)$ with respect to $x$ is $\frac{1}{\sqrt{1 - x^{2}}}$ .

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

Step 2

Find the second derivative.

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Step 2.1

Apply basic rules of exponents.

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Step 2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 - x^{2}}$ as ${(1 - x^{2})}^{\frac{1}{2}}$ .

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

Step 2.1.2

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

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

Step 2.1.3

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

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Step 2.1.3.1

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

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

Step 2.1.3.2

Combine $\frac{1}{2}$ and $- 1$ .

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

Step 2.1.3.3

Move the negative in front of the fraction.

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.3

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

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

Step 2.4

Combine $- 1$ and $\frac{2}{2}$ .

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $2$ .

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

Step 2.6.2

Subtract $2$ from $- 1$ .

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

Step 2.7

Combine fractions.

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Step 2.7.1

Move the negative in front of the fraction.

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

Step 2.7.2

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

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

Step 2.7.3

Move ${(1 - x^{2})}^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

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

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

Step 2.11

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

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

Step 2.12

Multiply.

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Step 2.12.1

Multiply $- 1$ by $- 1$ .

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

Step 2.12.2

Multiply $\frac{1}{2 {(1 - x^{2})}^{\frac{3}{2}}}$ by $1$ .

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

Step 2.13

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

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

Step 2.14

Simplify terms.

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Step 2.14.1

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

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

Step 2.14.2

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

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

Step 2.14.3

Cancel the common factor.

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

Step 2.14.4

Rewrite the expression.

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

Step 2.14.5

Reorder terms.

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