Calculus Examples

$f (x) = \arctan (\sqrt{x^{3}})$

Step 1

Rewrite $\frac{d}{d x} [\arctan (\sqrt{x^{3}})]$ as $\frac{d}{d x} [\arctan (x \sqrt{x})]$ .

Tap for more steps...

Step 1.1

Factor out $x^{2}$ .

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

Step 1.2

Pull terms out from under the radical.

$\frac{d}{d x} [\arctan (x \sqrt{x})]$

Step 2

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

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

Step 3

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

Tap for more steps...

Step 3.1

Multiply $x$ by $x^{\frac{1}{2}}$ .

Tap for more steps...

Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.2

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

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

Step 3.3

Combine the numerators over the common denominator.

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

Step 3.4

Add $2$ and $1$ .

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

Step 4

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

Tap for more steps...

Step 4.1

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

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

Step 4.2

The derivative of $\arctan (u)$ with respect to $u$ is $\frac{1}{1 + u^{2}}$ .

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

Step 4.3

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

Tap for more steps...

Step 4.3.1

Multiply $x$ by $x^{\frac{1}{2}}$ .

Tap for more steps...

Step 4.3.1.1

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

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

Step 4.3.1.2

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

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

Step 4.3.2

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

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

Step 4.3.3

Combine the numerators over the common denominator.

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

Step 4.3.4

Add $2$ and $1$ .

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

Step 5

Differentiate using the Power Rule.

Tap for more steps...

Step 5.1

Multiply the exponents in ${(x^{\frac{3}{2}})}^{2}$ .

Tap for more steps...

Step 5.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 5.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.1.2.1

Cancel the common factor.

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

Step 5.1.2.2

Rewrite the expression.

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

Step 5.2

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

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

Step 6

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

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

Step 7

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

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

Step 8

Combine the numerators over the common denominator.

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

Step 9

Simplify the numerator.

Tap for more steps...

Step 9.1

Multiply $- 1$ by $2$ .

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

Step 9.2

Subtract $2$ from $3$ .

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

Step 10

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

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

Step 11

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

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

Step 12

Move $2$ to the left of $1 + x^{3}$ .

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

Step 13

Simplify.

Tap for more steps...

Step 13.1

Apply the distributive property.

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

Step 13.2

Multiply $2$ by $1$ .

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

Step 13.3

Reorder terms.

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