Calculus Examples

$f (x) = 22 \arctan (\sqrt{s})$

Step 1

Differentiate using the Constant Multiple Rule.

Tap for more steps...

Step 1.1

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

$\frac{d}{d s} [22 \arctan (s^{\frac{1}{2}})]$

Step 1.2

Since $22$ is constant with respect to $s$ , the derivative of $22 \arctan (s^{\frac{1}{2}})$ with respect to $s$ is $22 \frac{d}{d s} [\arctan (s^{\frac{1}{2}})]$ .

$22 \frac{d}{d s} [\arctan (s^{\frac{1}{2}})]$

Step 2

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

Tap for more steps...

Step 2.1

To apply the Chain Rule, set $u$ as $s^{\frac{1}{2}}$ .

$22 (\frac{d}{d u} [\arctan (u)] \frac{d}{d s} [s^{\frac{1}{2}}])$

Step 2.2

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

$22 (\frac{1}{1 + u^{2}} \frac{d}{d s} [s^{\frac{1}{2}}])$

Step 2.3

Replace all occurrences of $u$ with $s^{\frac{1}{2}}$ .

$22 (\frac{1}{1 + {(s^{\frac{1}{2}})}^{2}} \frac{d}{d s} [s^{\frac{1}{2}}])$

Step 3

Multiply the exponents in ${(s^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 3.1

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

$22 (\frac{1}{1 + s^{\frac{1}{2} \cdot 2}} \frac{d}{d s} [s^{\frac{1}{2}}])$

Step 3.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.1

Cancel the common factor.

$22 (\frac{1}{1 + s^{\frac{1}{2} \cdot 2}} \frac{d}{d s} [s^{\frac{1}{2}}])$

Step 3.2.2

Rewrite the expression.

$22 (\frac{1}{1 + s^{1}} \frac{d}{d s} [s^{\frac{1}{2}}])$

Step 4

Simplify.

$22 (\frac{1}{1 + s} \frac{d}{d s} [s^{\frac{1}{2}}])$

Step 5

Combine $\frac{1}{1 + s}$ and $22$ .

$\frac{22}{1 + s} \frac{d}{d s} [s^{\frac{1}{2}}]$

Step 6

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

$\frac{22}{1 + s} (\frac{1}{2} s^{\frac{1}{2} - 1})$

Step 7

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

$\frac{22}{1 + s} (\frac{1}{2} s^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 8

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

$\frac{22}{1 + s} (\frac{1}{2} s^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 9

Combine the numerators over the common denominator.

$\frac{22}{1 + s} (\frac{1}{2} s^{\frac{1 - 1 \cdot 2}{2}})$

Step 10

Simplify the numerator.

Tap for more steps...

Step 10.1

Multiply $- 1$ by $2$ .

$\frac{22}{1 + s} (\frac{1}{2} s^{\frac{1 - 2}{2}})$

Step 10.2

Subtract $2$ from $1$ .

$\frac{22}{1 + s} (\frac{1}{2} s^{\frac{- 1}{2}})$

Step 11

Move the negative in front of the fraction.

$\frac{22}{1 + s} (\frac{1}{2} s^{- \frac{1}{2}})$

Step 12

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

$\frac{22}{1 + s} \cdot \frac{s^{- \frac{1}{2}}}{2}$

Step 13

Multiply $\frac{22}{1 + s}$ by $\frac{s^{- \frac{1}{2}}}{2}$ .

$\frac{22 s^{- \frac{1}{2}}}{(1 + s) \cdot 2}$

Step 14

Simplify the expression.

Tap for more steps...

Step 14.1

Move $2$ to the left of $1 + s$ .

$\frac{22 s^{- \frac{1}{2}}}{2 \cdot (1 + s)}$

Step 14.2

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

$\frac{22}{2 (1 + s) s^{\frac{1}{2}}}$

Step 15

Factor $2$ out of $22$ .

$\frac{2 \cdot 11}{2 (1 + s) s^{\frac{1}{2}}}$

Step 16

Cancel the common factors.

Tap for more steps...

Step 16.1

Factor $2$ out of $2 (1 + s) s^{\frac{1}{2}}$ .

$\frac{2 \cdot 11}{2 ((1 + s) s^{\frac{1}{2}})}$

Step 16.2

Cancel the common factor.

$\frac{2 \cdot 11}{2 ((1 + s) s^{\frac{1}{2}})}$

Step 16.3

Rewrite the expression.

$\frac{11}{(1 + s) s^{\frac{1}{2}}}$

Step 17

Simplify.

Tap for more steps...

Step 17.1

Apply the distributive property.

$\frac{11}{1 s^{\frac{1}{2}} + s \cdot s^{\frac{1}{2}}}$

Step 17.2

Combine terms.

Tap for more steps...

Step 17.2.1

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

$\frac{11}{s^{\frac{1}{2}} + s \cdot s^{\frac{1}{2}}}$

Step 17.2.2

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

Tap for more steps...

Step 17.2.2.1

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

Tap for more steps...

Step 17.2.2.1.1

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

$\frac{11}{s^{\frac{1}{2}} + s^{1} s^{\frac{1}{2}}}$

Step 17.2.2.1.2

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

$\frac{11}{s^{\frac{1}{2}} + s^{1 + \frac{1}{2}}}$

Step 17.2.2.2

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

$\frac{11}{s^{\frac{1}{2}} + s^{\frac{2}{2} + \frac{1}{2}}}$

Step 17.2.2.3

Combine the numerators over the common denominator.

$\frac{11}{s^{\frac{1}{2}} + s^{\frac{2 + 1}{2}}}$

Step 17.2.2.4

Add $2$ and $1$ .

$\frac{11}{s^{\frac{1}{2}} + s^{\frac{3}{2}}}$

Step 17.3

Simplify the denominator.

Tap for more steps...

Step 17.3.1

Factor $s^{\frac{1}{2}}$ out of $s^{\frac{1}{2}} + s^{\frac{3}{2}}$ .

Tap for more steps...

Step 17.3.1.1

Multiply by $1$ .

$\frac{11}{s^{\frac{1}{2}} \cdot 1 + s^{\frac{3}{2}}}$

Step 17.3.1.2

Factor $s^{\frac{1}{2}}$ out of $s^{\frac{3}{2}}$ .

$\frac{11}{s^{\frac{1}{2}} \cdot 1 + s^{\frac{1}{2}} s^{\frac{2}{2}}}$

Step 17.3.1.3

Factor $s^{\frac{1}{2}}$ out of $s^{\frac{1}{2}} \cdot 1 + s^{\frac{1}{2}} s^{\frac{2}{2}}$ .

$\frac{11}{s^{\frac{1}{2}} (1 + s^{\frac{2}{2}})}$

Step 17.3.2

Divide $2$ by $2$ .

$\frac{11}{s^{\frac{1}{2}} (1 + s^{1})}$

Step 17.3.3

Simplify.

$\frac{11}{s^{\frac{1}{2}} (1 + s)}$