Calculus Examples

$y = 6 \sqrt{4 + x}$

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

$\frac{d}{d x} [6 {(4 + x)}^{\frac{1}{2}}]$

Step 1.2

Since $6$ is constant with respect to $x$ , the derivative of $6 {(4 + x)}^{\frac{1}{2}}$ with respect to $x$ is $6 \frac{d}{d x} [{(4 + x)}^{\frac{1}{2}}]$ .

$6 \frac{d}{d x} [{(4 + x)}^{\frac{1}{2}}]$

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 $g (x) = 4 + x$ .

Tap for more steps...

Step 2.1

To apply the Chain Rule, set $u$ as $4 + x$ .

$6 (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [4 + x])$

Step 2.2

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

$6 (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [4 + x])$

Step 2.3

Replace all occurrences of $u$ with $4 + x$ .

$6 (\frac{1}{2} {(4 + x)}^{\frac{1}{2} - 1} \frac{d}{d x} [4 + x])$

Step 3

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

$6 (\frac{1}{2} {(4 + x)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [4 + x])$

Step 4

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

$6 (\frac{1}{2} {(4 + x)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [4 + x])$

Step 5

Combine the numerators over the common denominator.

$6 (\frac{1}{2} {(4 + x)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [4 + x])$

Step 6

Simplify the numerator.

Tap for more steps...

Step 6.1

Multiply $- 1$ by $2$ .

$6 (\frac{1}{2} {(4 + x)}^{\frac{1 - 2}{2}} \frac{d}{d x} [4 + x])$

Step 6.2

Subtract $2$ from $1$ .

$6 (\frac{1}{2} {(4 + x)}^{\frac{- 1}{2}} \frac{d}{d x} [4 + x])$

Step 7

Move the negative in front of the fraction.

$6 (\frac{1}{2} {(4 + x)}^{- \frac{1}{2}} \frac{d}{d x} [4 + x])$

Step 8

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

$6 (\frac{{(4 + x)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [4 + x])$

Step 9

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

$6 (\frac{1}{2 {(4 + x)}^{\frac{1}{2}}} \frac{d}{d x} [4 + x])$

Step 10

Combine $\frac{1}{2 {(4 + x)}^{\frac{1}{2}}}$ and $6$ .

$\frac{6}{2 {(4 + x)}^{\frac{1}{2}}} \frac{d}{d x} [4 + x]$

Step 11

Factor $2$ out of $6$ .

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

Step 12

Cancel the common factors.

Tap for more steps...

Step 12.1

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

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

Step 12.2

Cancel the common factor.

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

Step 12.3

Rewrite the expression.

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

Step 13

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

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

Step 14

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$\frac{3}{{(4 + x)}^{\frac{1}{2}}} (0 + \frac{d}{d x} [x])$

Step 15

Add $0$ and $\frac{d}{d x} [x]$ .

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

Step 16

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

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

Step 17

Simplify the expression.

Tap for more steps...

Step 17.1

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

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

Step 17.2

Reorder terms.

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