Calculus Examples

$y = x^{3} \sqrt{x + 1}$

Step 1

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

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

Step 2

Differentiate both sides of the equation.

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

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

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

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 $g (x) = x + 1$ .

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Step 4.2.1

To apply the Chain Rule, set $u$ as $x + 1$ .

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

Step 4.2.2

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

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

Step 4.2.3

Replace all occurrences of $u$ with $x + 1$ .

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Combine the numerators over the common denominator.

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

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 1$ by $2$ .

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

Step 4.6.2

Subtract $2$ from $1$ .

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

Step 4.7

Combine fractions.

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Step 4.7.1

Move the negative in front of the fraction.

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

Step 4.7.2

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

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

Step 4.7.3

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

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

Step 4.7.4

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

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

Step 4.8

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

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

Step 4.9

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

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

Step 4.10

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

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

Step 4.11

Simplify the expression.

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Step 4.11.1

Add $1$ and $0$ .

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

Step 4.11.2

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

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

Step 4.12

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

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

Step 4.13

Move $3$ to the left of ${(x + 1)}^{\frac{1}{2}}$ .

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

Step 4.14

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

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Step 4.14.1

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

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

Step 4.14.2

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

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

Step 4.14.3

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

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

Step 4.14.4

Combine the numerators over the common denominator.

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

Step 4.15

Multiply $2$ by $3$ .

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

Step 4.16

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

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Step 4.16.1

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

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

Step 4.16.2

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

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

Step 4.16.3

Combine the numerators over the common denominator.

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

Step 4.16.4

Add $1$ and $1$ .

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

Step 4.16.5

Divide $2$ by $2$ .

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

Step 4.17

Simplify $6 x^{2} {(x + 1)}^{1}$ .

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

Step 4.18

Simplify.

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Step 4.18.1

Apply the distributive property.

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

Step 4.18.2

Simplify the numerator.

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Step 4.18.2.1

Simplify each term.

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Step 4.18.2.1.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 4.18.2.1.1.1

Move $x$ .

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

Step 4.18.2.1.1.2

Multiply $x$ by $x^{2}$ .

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Step 4.18.2.1.1.2.1

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

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

Step 4.18.2.1.1.2.2

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

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

Step 4.18.2.1.1.3

Add $1$ and $2$ .

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

Step 4.18.2.1.2

Multiply $6$ by $1$ .

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

Step 4.18.2.2

Add $x^{3}$ and $6 x^{3}$ .

$\frac{7 x^{3} + 6 x^{2}}{2 {(x + 1)}^{\frac{1}{2}}}$

Step 4.18.3

Factor $x^{2}$ out of $7 x^{3} + 6 x^{2}$ .

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Step 4.18.3.1

Factor $x^{2}$ out of $7 x^{3}$ .

$\frac{x^{2} (7 x) + 6 x^{2}}{2 {(x + 1)}^{\frac{1}{2}}}$

Step 4.18.3.2

Factor $x^{2}$ out of $6 x^{2}$ .

$\frac{x^{2} (7 x) + x^{2} \cdot 6}{2 {(x + 1)}^{\frac{1}{2}}}$

Step 4.18.3.3

Factor $x^{2}$ out of $x^{2} (7 x) + x^{2} \cdot 6$ .

$\frac{x^{2} (7 x + 6)}{2 {(x + 1)}^{\frac{1}{2}}}$

Step 5

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

$y' = \frac{x^{2} (7 x + 6)}{2 {(x + 1)}^{\frac{1}{2}}}$

Step 6

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

$\frac{d y}{d x} = \frac{x^{2} (7 x + 6)}{2 {(x + 1)}^{\frac{1}{2}}}$