Calculus Examples

$\frac{d}{d x} (\frac{1}{3} \sqrt{{(x^{2} + y^{2})}^{3}})$

Step 1

Rewrite $\frac{d}{d x} [\frac{1}{3} \sqrt{{(x^{2} + y^{2})}^{3}}]$ as $\frac{d}{d x} [\frac{1}{3} ((x^{2} + y^{2}) \sqrt{x^{2} + y^{2}})]$ .

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Step 1.1

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

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.2

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

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

Step 3.3

Combine the numerators over the common denominator.

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

Step 3.4

Add $2$ and $1$ .

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

Step 4

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{1}{3} {(x^{2} + y^{2})}^{\frac{3}{2}}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [{(x^{2} + y^{2})}^{\frac{3}{2}}]$ .

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

Step 5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{3}{2}}$ and $g (x) = x^{2} + y^{2}$ .

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Step 5.1

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

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

Step 5.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{3}{2}$ .

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

Step 5.3

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

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

Step 6

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

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

Step 7

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

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

Step 8

Combine the numerators over the common denominator.

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

Step 9

Simplify the numerator.

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Step 9.1

Multiply $- 1$ by $2$ .

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

Step 9.2

Subtract $2$ from $3$ .

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

Step 10

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

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

Step 11

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

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

Step 12

Multiply $2$ by $3$ .

$\frac{3 {(x^{2} + y^{2})}^{\frac{1}{2}}}{6} \frac{d}{d x} [x^{2} + y^{2}]$

Step 13

Factor $3$ out of $3 {(x^{2} + y^{2})}^{\frac{1}{2}}$ .

$\frac{3 ({(x^{2} + y^{2})}^{\frac{1}{2}})}{6} \frac{d}{d x} [x^{2} + y^{2}]$

Step 14

Cancel the common factors.

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Step 14.1

Factor $3$ out of $6$ .

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

Step 14.2

Cancel the common factor.

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

Step 14.3

Rewrite the expression.

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

Step 15

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

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

Step 16

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

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

Step 17

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

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

Step 18

Simplify terms.

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Step 18.1

Add $2 x$ and $0$ .

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

Step 18.2

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

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

Step 18.3

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

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

Step 18.4

Cancel the common factor.

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

Step 18.5

Simplify the expression.

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Step 18.5.1

Divide ${(x^{2} + y^{2})}^{\frac{1}{2}} x$ by $1$ .

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

Step 18.5.2

Reorder the factors of ${(x^{2} + y^{2})}^{\frac{1}{2}} x$ .

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