Calculus Examples

$\sqrt{x^{3} + 3 x^{2}}$

Step 1

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

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Step 1.1

Rewrite $x^{3} + 3 x^{2}$ as $x^{2} (x + 3)$ .

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Step 1.1.1

Factor $x^{2}$ out of $x^{3}$ .

$\frac{d}{d x} [\sqrt{x^{2} x + 3 x^{2}}]$

Step 1.1.2

Factor $x^{2}$ out of $3 x^{2}$ .

$\frac{d}{d x} [\sqrt{x^{2} x + x^{2} \cdot 3}]$

Step 1.1.3

Factor $x^{2}$ out of $x^{2} x + x^{2} \cdot 3$ .

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

Step 1.2

Pull terms out from under the radical.

$\frac{d}{d x} [x \sqrt{x + 3}]$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 3}$ as ${(x + 3)}^{\frac{1}{2}}$ .

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

Step 3

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 $g (x) = {(x + 3)}^{\frac{1}{2}}$ .

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

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = x + 3$ .

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Step 4.1

To apply the Chain Rule, set $u$ as $x + 3$ .

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

Step 4.2

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

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

Step 4.3

Replace all occurrences of $u$ with $x + 3$ .

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

Step 5

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

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

Step 6

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

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

Step 7

Combine the numerators over the common denominator.

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

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $2$ .

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

Step 8.2

Subtract $2$ from $1$ .

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

Step 9

Combine fractions.

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Step 9.1

Move the negative in front of the fraction.

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

Step 9.2

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

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

Step 9.3

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

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

Step 9.4

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

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

Step 10

By the Sum Rule, the derivative of $x + 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [3]$ .

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

Step 11

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

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

Step 12

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

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

Step 13

Simplify the expression.

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Step 13.1

Add $1$ and $0$ .

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

Step 13.2

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

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

Step 14

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

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

Step 15

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

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

Step 16

To write ${(x + 3)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2 {(x + 3)}^{\frac{1}{2}}}{2 {(x + 3)}^{\frac{1}{2}}}$ .

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

Step 17

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

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

Step 18

Combine the numerators over the common denominator.

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

Step 19

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

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Step 19.1

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

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

Step 19.2

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

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

Step 19.3

Combine the numerators over the common denominator.

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

Step 19.4

Add $1$ and $1$ .

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

Step 19.5

Divide $2$ by $2$ .

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

Step 20

Simplify ${(x + 3)}^{1} \cdot 2$ .

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

Step 21

Move $2$ to the left of $x + 3$ .

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

Step 22

Simplify.

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Step 22.1

Apply the distributive property.

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

Step 22.2

Simplify the numerator.

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Step 22.2.1

Multiply $2$ by $3$ .

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

Step 22.2.2

Add $x$ and $2 x$ .

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

Step 22.3

Factor $3$ out of $3 x + 6$ .

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Step 22.3.1

Factor $3$ out of $3 x$ .

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

Step 22.3.2

Factor $3$ out of $6$ .

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

Step 22.3.3

Factor $3$ out of $3 x + 3 \cdot 2$ .

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