Calculus Examples

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

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

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

$3 x + 3 \cdot 2$ .

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