Calculus Examples

$\sqrt{4 \sin (x) + 2}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 \sin (x) + 2}$ as ${(4 \sin (x) + 2)}^{\frac{1}{2}}$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 4 \sin (x) + 2$ .

Tap for more steps...

Step 2.1

To apply the Chain Rule, set $u$ as $4 \sin (x) + 2$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $4 \sin (x) + 2$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

Tap for more steps...

Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $1$ .

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

Step 7

Differentiate.

Tap for more steps...

Step 7.1

Move the negative in front of the fraction.

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

Step 7.2

Combine fractions.

Tap for more steps...

Step 7.2.1

Combine $\frac{1}{2}$ and ${(4 \sin (x) + 2)}^{- \frac{1}{2}}$ .

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

Step 7.2.2

Move ${(4 \sin (x) + 2)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 7.3

By the Sum Rule, the derivative of $4 \sin (x) + 2$ with respect to $x$ is $\frac{d}{d x} [4 \sin (x)] + \frac{d}{d x} [2]$ .

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

Step 7.4

Since $4$ is constant with respect to $x$ , the derivative of $4 \sin (x)$ with respect to $x$ is $4 \frac{d}{d x} [\sin (x)]$ .

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

Step 8

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

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

Step 9

Differentiate using the Constant Rule.

Tap for more steps...

Step 9.1

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

$\frac{1}{2 {(4 \sin (x) + 2)}^{\frac{1}{2}}} (4 \cos (x) + 0)$

Step 9.2

Simplify terms.

Tap for more steps...

Step 9.2.1

Add $4 \cos (x)$ and $0$ .

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

Step 9.2.2

Combine $4$ and $\frac{1}{2 {(4 \sin (x) + 2)}^{\frac{1}{2}}}$ .

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

Step 9.2.3

Combine $\frac{4}{2 {(4 \sin (x) + 2)}^{\frac{1}{2}}}$ and $\cos (x)$ .

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

Step 9.2.4

Factor $2$ out of $4 \cos (x)$ .

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

Step 10

Cancel the common factors.

Tap for more steps...

Step 10.1

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

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

Step 10.2

Cancel the common factor.

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

Step 10.3

Rewrite the expression.

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

Step 11

Reorder terms.

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