Calculus Examples

$\ln (\sqrt{x} + 2)$

Step 1

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

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

Step 2

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

Tap for more steps...

Step 2.1

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

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

Step 2.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

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

Step 2.3

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

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

Step 3

Differentiate.

Tap for more steps...

Step 3.1

By the Sum Rule, the derivative of $x^{\frac{1}{2}} + 2$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [2]$ .

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

Step 3.2

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

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

Tap for more steps...

Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Combine fractions.

Tap for more steps...

Step 8.1

Move the negative in front of the fraction.

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

Step 8.2

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

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

Step 8.3

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

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

Step 9

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$\frac{1}{x^{\frac{1}{2}} + 2} (\frac{1}{2 x^{\frac{1}{2}}} + 0)$

Step 10

Combine fractions.

Tap for more steps...

Step 10.1

Add $\frac{1}{2 x^{\frac{1}{2}}}$ and $0$ .

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

Step 10.2

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

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

Step 10.3

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

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

Step 11

Simplify.

Tap for more steps...

Step 11.1

Apply the distributive property.

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

Step 11.2

Apply the distributive property.

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

Step 11.3

Combine terms.

Tap for more steps...

Step 11.3.1

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

Tap for more steps...

Step 11.3.1.1

Move $x^{\frac{1}{2}}$ .

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

Step 11.3.1.2

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

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

Step 11.3.1.3

Combine the numerators over the common denominator.

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

Step 11.3.1.4

Add $1$ and $1$ .

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

Step 11.3.1.5

Divide $2$ by $2$ .

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

Step 11.3.2

Simplify $2 x^{1}$ .

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

Step 11.3.3

Multiply $2$ by $2$ .

$\frac{1}{2 x + 4 x^{\frac{1}{2}}}$

Step 11.4

Factor $2 x^{\frac{1}{2}}$ out of $2 x + 4 x^{\frac{1}{2}}$ .

Tap for more steps...

Step 11.4.1

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

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

Step 11.4.2

Factor $2 x^{\frac{1}{2}}$ out of $4 x^{\frac{1}{2}}$ .

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

Step 11.4.3

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

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