Calculus Examples

$y = x \sqrt{1 - x^{2}}$

Step 1

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

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

Step 2

Differentiate both sides of the equation.

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

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

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

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

Step 4.2

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 4.2.1

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

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

Step 4.2.2

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

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

Step 4.2.3

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Combine the numerators over the common denominator.

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

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 1$ by $2$ .

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

Step 4.6.2

Subtract $2$ from $1$ .

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

Step 4.7

Combine fractions.

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Step 4.7.1

Move the negative in front of the fraction.

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

Step 4.7.2

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

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

Step 4.7.3

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

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

Step 4.7.4

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

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

Step 4.8

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

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

Step 4.9

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

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

Step 4.10

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

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

Step 4.11

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

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

Step 4.12

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

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

Step 4.13

Combine fractions.

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Step 4.13.1

Multiply $2$ by $- 1$ .

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

Step 4.13.2

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

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

Step 4.13.3

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

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

Step 4.14

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

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

Step 4.15

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

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

Step 4.16

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

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

Step 4.17

Add $1$ and $1$ .

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

Step 4.18

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

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

Step 4.19

Cancel the common factors.

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Step 4.19.1

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

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

Step 4.19.2

Cancel the common factor.

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

Step 4.19.3

Rewrite the expression.

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

Step 4.20

Move the negative in front of the fraction.

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

Step 4.21

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

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

Step 4.22

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

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

Step 4.23

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

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

Step 4.24

Combine the numerators over the common denominator.

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

Step 4.25

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

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Step 4.25.1

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

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

Step 4.25.2

Combine the numerators over the common denominator.

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

Step 4.25.3

Add $1$ and $1$ .

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

Step 4.25.4

Divide $2$ by $2$ .

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

Step 4.26

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

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

Step 4.27

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

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

Step 4.28

Reorder terms.

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

Step 5

Reform the equation by setting the left side equal to the right side.

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

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

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