Calculus Examples

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

Step 1

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

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

Step 2

Differentiate both sides of the equation.

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

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

Tap for more steps...

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

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

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

Tap for more steps...

Step 4.2.1

To apply the Chain Rule, set $u$ as $x^{2} + 3$ .

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

Step 4.2.2

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

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

Step 4.2.3

Replace all occurrences of $u$ with $x^{2} + 3$ .

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Combine the numerators over the common denominator.

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

Step 4.6

Simplify the numerator.

Tap for more steps...

Step 4.6.1

Multiply $- 1$ by $2$ .

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

Step 4.6.2

Subtract $2$ from $1$ .

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

Step 4.7

Combine fractions.

Tap for more steps...

Step 4.7.1

Move the negative in front of the fraction.

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

Step 4.7.2

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

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

Step 4.7.3

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

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

Step 4.7.4

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

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

Step 4.8

By the Sum Rule, the derivative of $x^{2} + 3$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3]$ .

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

Step 4.9

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

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

Step 4.10

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

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

Step 4.11

Combine fractions.

Tap for more steps...

Step 4.11.1

Add $2 x$ and $0$ .

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

Step 4.11.2

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

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

Step 4.11.3

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

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

Step 4.12

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

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

Step 4.13

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

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

Step 4.14

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 4.14.1

Add $1$ and $3$ .

$\frac{2 x^{4}}{2 {(x^{2} + 3)}^{\frac{1}{2}}} + {(x^{2} + 3)}^{\frac{1}{2}} \frac{d}{d x} [x^{3}]$

Step 4.14.2

Cancel the common factor.

$\frac{2 x^{4}}{2 {(x^{2} + 3)}^{\frac{1}{2}}} + {(x^{2} + 3)}^{\frac{1}{2}} \frac{d}{d x} [x^{3}]$

Step 4.14.3

Rewrite the expression.

$\frac{x^{4}}{{(x^{2} + 3)}^{\frac{1}{2}}} + {(x^{2} + 3)}^{\frac{1}{2}} \frac{d}{d x} [x^{3}]$

Step 4.15

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

$\frac{x^{4}}{{(x^{2} + 3)}^{\frac{1}{2}}} + {(x^{2} + 3)}^{\frac{1}{2}} (3 x^{2})$

Step 4.16

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

$\frac{x^{4}}{{(x^{2} + 3)}^{\frac{1}{2}}} + 3 \cdot {(x^{2} + 3)}^{\frac{1}{2}} x^{2}$

Step 4.17

Combine $\frac{x^{4}}{{(x^{2} + 3)}^{\frac{1}{2}}}$ and $3 {(x^{2} + 3)}^{\frac{1}{2}} x^{2}$ using a common denominator.

Tap for more steps...

Step 4.17.1

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

$\frac{x^{4}}{{(x^{2} + 3)}^{\frac{1}{2}}} + 3 x^{2} {(x^{2} + 3)}^{\frac{1}{2}}$

Step 4.17.2

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

$\frac{x^{4}}{{(x^{2} + 3)}^{\frac{1}{2}}} + \frac{3 x^{2} {(x^{2} + 3)}^{\frac{1}{2}} {(x^{2} + 3)}^{\frac{1}{2}}}{{(x^{2} + 3)}^{\frac{1}{2}}}$

Step 4.17.3

Combine the numerators over the common denominator.

$\frac{x^{4} + 3 x^{2} {(x^{2} + 3)}^{\frac{1}{2}} {(x^{2} + 3)}^{\frac{1}{2}}}{{(x^{2} + 3)}^{\frac{1}{2}}}$

Step 4.18

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

Tap for more steps...

Step 4.18.1

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

$\frac{x^{4} + 3 x^{2} ({(x^{2} + 3)}^{\frac{1}{2}} {(x^{2} + 3)}^{\frac{1}{2}})}{{(x^{2} + 3)}^{\frac{1}{2}}}$

Step 4.18.2

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

$\frac{x^{4} + 3 x^{2} {(x^{2} + 3)}^{\frac{1}{2} + \frac{1}{2}}}{{(x^{2} + 3)}^{\frac{1}{2}}}$

Step 4.18.3

Combine the numerators over the common denominator.

$\frac{x^{4} + 3 x^{2} {(x^{2} + 3)}^{\frac{1 + 1}{2}}}{{(x^{2} + 3)}^{\frac{1}{2}}}$

Step 4.18.4

Add $1$ and $1$ .

$\frac{x^{4} + 3 x^{2} {(x^{2} + 3)}^{\frac{2}{2}}}{{(x^{2} + 3)}^{\frac{1}{2}}}$

Step 4.18.5

Divide $2$ by $2$ .

$\frac{x^{4} + 3 x^{2} {(x^{2} + 3)}^{1}}{{(x^{2} + 3)}^{\frac{1}{2}}}$

Step 4.19

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

$\frac{x^{4} + 3 x^{2} (x^{2} + 3)}{{(x^{2} + 3)}^{\frac{1}{2}}}$

Step 4.20

Simplify.

Tap for more steps...

Step 4.20.1

Apply the distributive property.

$\frac{x^{4} + 3 x^{2} x^{2} + 3 x^{2} \cdot 3}{{(x^{2} + 3)}^{\frac{1}{2}}}$

Step 4.20.2

Simplify the numerator.

Tap for more steps...

Step 4.20.2.1

Simplify each term.

Tap for more steps...

Step 4.20.2.1.1

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

Tap for more steps...

Step 4.20.2.1.1.1

Move $x^{2}$ .

$\frac{x^{4} + 3 (x^{2} x^{2}) + 3 x^{2} \cdot 3}{{(x^{2} + 3)}^{\frac{1}{2}}}$

Step 4.20.2.1.1.2

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

$\frac{x^{4} + 3 x^{2 + 2} + 3 x^{2} \cdot 3}{{(x^{2} + 3)}^{\frac{1}{2}}}$

Step 4.20.2.1.1.3

Add $2$ and $2$ .

$\frac{x^{4} + 3 x^{4} + 3 x^{2} \cdot 3}{{(x^{2} + 3)}^{\frac{1}{2}}}$

Step 4.20.2.1.2

Multiply $3$ by $3$ .

$\frac{x^{4} + 3 x^{4} + 9 x^{2}}{{(x^{2} + 3)}^{\frac{1}{2}}}$

Step 4.20.2.2

Add $x^{4}$ and $3 x^{4}$ .

$\frac{4 x^{4} + 9 x^{2}}{{(x^{2} + 3)}^{\frac{1}{2}}}$

Step 4.20.3

Factor $x^{2}$ out of $4 x^{4} + 9 x^{2}$ .

Tap for more steps...

Step 4.20.3.1

Factor $x^{2}$ out of $4 x^{4}$ .

$\frac{x^{2} (4 x^{2}) + 9 x^{2}}{{(x^{2} + 3)}^{\frac{1}{2}}}$

Step 4.20.3.2

Factor $x^{2}$ out of $9 x^{2}$ .

$\frac{x^{2} (4 x^{2}) + x^{2} \cdot 9}{{(x^{2} + 3)}^{\frac{1}{2}}}$

Step 4.20.3.3

Factor $x^{2}$ out of $x^{2} (4 x^{2}) + x^{2} \cdot 9$ .

$\frac{x^{2} (4 x^{2} + 9)}{{(x^{2} + 3)}^{\frac{1}{2}}}$

Step 5

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

$y' = \frac{x^{2} (4 x^{2} + 9)}{{(x^{2} + 3)}^{\frac{1}{2}}}$

Step 6

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

$\frac{d y}{d x} = \frac{x^{2} (4 x^{2} + 9)}{{(x^{2} + 3)}^{\frac{1}{2}}}$