Calculus Examples

$f (x) = {(6 x - 2)}^{\frac{1}{2}}$

Step 1

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 $g (x) = 6 x - 2$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $6 x - 2$ .

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [6 x - 2]$

Step 1.2

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

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [6 x - 2]$

Step 1.3

Replace all occurrences of $u$ with $6 x - 2$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Multiply $- 1$ by $2$ .

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

Step 5.2

Subtract $2$ from $1$ .

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

Step 6

Combine fractions.

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Step 6.1

Move the negative in front of the fraction.

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

Step 6.2

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

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

Step 6.3

Move ${(6 x - 2)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 7

By the Sum Rule, the derivative of $6 x - 2$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [- 2]$ .

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

Step 8

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

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

Step 9

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

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

Step 10

Multiply $6$ by $1$ .

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

Step 11

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

$\frac{1}{2 {(6 x - 2)}^{\frac{1}{2}}} (6 + 0)$

Step 12

Simplify terms.

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Step 12.1

Add $6$ and $0$ .

$\frac{1}{2 {(6 x - 2)}^{\frac{1}{2}}} \cdot 6$

Step 12.2

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

$\frac{6}{2 {(6 x - 2)}^{\frac{1}{2}}}$

Step 12.3

Factor $2$ out of $6$ .

$\frac{2 \cdot 3}{2 {(6 x - 2)}^{\frac{1}{2}}}$

Step 13

Cancel the common factors.

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Step 13.1

Factor $2$ out of $2 {(6 x - 2)}^{\frac{1}{2}}$ .

$\frac{2 \cdot 3}{2 ({(6 x - 2)}^{\frac{1}{2}})}$

Step 13.2

Cancel the common factor.

$\frac{2 \cdot 3}{2 {(6 x - 2)}^{\frac{1}{2}}}$

Step 13.3

Rewrite the expression.

$\frac{3}{{(6 x - 2)}^{\frac{1}{2}}}$