Calculus Examples

$\sqrt{4 - x^{2} - y^{2}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 - x^{2} - y^{2}}$ as ${(4 - x^{2} - y^{2})}^{\frac{1}{2}}$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 4 - x^{2} - y^{2}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $4 - x^{2} - y^{2}$ .

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [4 - x^{2} - y^{2}]$

Step 2.2

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

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [4 - x^{2} - y^{2}]$

Step 2.3

Replace all occurrences of $u$ with $4 - x^{2} - y^{2}$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $1$ .

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

Step 7

Combine fractions.

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Step 7.1

Move the negative in front of the fraction.

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

Step 7.2

Combine $\frac{1}{2}$ and ${(4 - x^{2} - y^{2})}^{- \frac{1}{2}}$ .

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

Step 7.3

Move ${(4 - x^{2} - y^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 8

By the Sum Rule, the derivative of $4 - x^{2} - y^{2}$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- y^{2}]$ .

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

Step 9

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

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

Step 10

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

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

Step 11

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

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

Step 12

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

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

Step 13

Multiply $2$ by $- 1$ .

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

Step 14

Since $- y^{2}$ is constant with respect to $x$ , the derivative of $- y^{2}$ with respect to $x$ is $0$ .

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

Step 15

Simplify terms.

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Step 15.1

Add $- 2 x$ and $0$ .

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

Step 15.2

Combine $- 2$ and $\frac{1}{2 {(4 - x^{2} - y^{2})}^{\frac{1}{2}}}$ .

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

Step 15.3

Combine $\frac{- 2}{2 {(4 - x^{2} - y^{2})}^{\frac{1}{2}}}$ and $x$ .

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

Step 15.4

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

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

Step 16

Cancel the common factors.

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Step 16.1

Factor $2$ out of $2 {(4 - x^{2} - y^{2})}^{\frac{1}{2}}$ .

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

Step 16.2

Cancel the common factor.

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

Step 16.3

Rewrite the expression.

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

Step 17

Move the negative in front of the fraction.

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