Calculus Examples

$f (x) = 4 \sqrt{6 x^{2} - 7 x}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 x^{2} - 7 x}$ as ${(6 x^{2} - 7 x)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [4 {(6 x^{2} - 7 x)}^{\frac{1}{2}}]$

Step 1.2

Since $4$ is constant with respect to $x$ , the derivative of $4 {(6 x^{2} - 7 x)}^{\frac{1}{2}}$ with respect to $x$ is $4 \frac{d}{d x} [{(6 x^{2} - 7 x)}^{\frac{1}{2}}]$ .

$4 \frac{d}{d x} [{(6 x^{2} - 7 x)}^{\frac{1}{2}}]$

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 $g (x) = 6 x^{2} - 7 x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $6 x^{2} - 7 x$ .

$4 (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [6 x^{2} - 7 x])$

Step 2.2

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

$4 (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [6 x^{2} - 7 x])$

Step 2.3

Replace all occurrences of $u$ with $6 x^{2} - 7 x$ .

$4 (\frac{1}{2} {(6 x^{2} - 7 x)}^{\frac{1}{2} - 1} \frac{d}{d x} [6 x^{2} - 7 x])$

Step 3

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

$4 (\frac{1}{2} {(6 x^{2} - 7 x)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [6 x^{2} - 7 x])$

Step 4

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

$4 (\frac{1}{2} {(6 x^{2} - 7 x)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [6 x^{2} - 7 x])$

Step 5

Combine the numerators over the common denominator.

$4 (\frac{1}{2} {(6 x^{2} - 7 x)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [6 x^{2} - 7 x])$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

$4 (\frac{1}{2} {(6 x^{2} - 7 x)}^{\frac{1 - 2}{2}} \frac{d}{d x} [6 x^{2} - 7 x])$

Step 6.2

Subtract $2$ from $1$ .

$4 (\frac{1}{2} {(6 x^{2} - 7 x)}^{\frac{- 1}{2}} \frac{d}{d x} [6 x^{2} - 7 x])$

Step 7

Move the negative in front of the fraction.

$4 (\frac{1}{2} {(6 x^{2} - 7 x)}^{- \frac{1}{2}} \frac{d}{d x} [6 x^{2} - 7 x])$

Step 8

Combine $\frac{1}{2}$ and ${(6 x^{2} - 7 x)}^{- \frac{1}{2}}$ .

$4 (\frac{{(6 x^{2} - 7 x)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [6 x^{2} - 7 x])$

Step 9

Move ${(6 x^{2} - 7 x)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$4 (\frac{1}{2 {(6 x^{2} - 7 x)}^{\frac{1}{2}}} \frac{d}{d x} [6 x^{2} - 7 x])$

Step 10

Combine $\frac{1}{2 {(6 x^{2} - 7 x)}^{\frac{1}{2}}}$ and $4$ .

$\frac{4}{2 {(6 x^{2} - 7 x)}^{\frac{1}{2}}} \frac{d}{d x} [6 x^{2} - 7 x]$

Step 11

Factor $2$ out of $4$ .

$\frac{2 \cdot 2}{2 {(6 x^{2} - 7 x)}^{\frac{1}{2}}} \frac{d}{d x} [6 x^{2} - 7 x]$

Step 12

Cancel the common factors.

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Step 12.1

Factor $2$ out of $2 {(6 x^{2} - 7 x)}^{\frac{1}{2}}$ .

$\frac{2 \cdot 2}{2 ({(6 x^{2} - 7 x)}^{\frac{1}{2}})} \frac{d}{d x} [6 x^{2} - 7 x]$

Step 12.2

Cancel the common factor.

$\frac{2 \cdot 2}{2 {(6 x^{2} - 7 x)}^{\frac{1}{2}}} \frac{d}{d x} [6 x^{2} - 7 x]$

Step 12.3

Rewrite the expression.

$\frac{2}{{(6 x^{2} - 7 x)}^{\frac{1}{2}}} \frac{d}{d x} [6 x^{2} - 7 x]$

Step 13

By the Sum Rule, the derivative of $6 x^{2} - 7 x$ with respect to $x$ is $\frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [- 7 x]$ .

$\frac{2}{{(6 x^{2} - 7 x)}^{\frac{1}{2}}} (\frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [- 7 x])$

Step 14

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

$\frac{2}{{(6 x^{2} - 7 x)}^{\frac{1}{2}}} (6 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 7 x])$

Step 15

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

$\frac{2}{{(6 x^{2} - 7 x)}^{\frac{1}{2}}} (6 (2 x) + \frac{d}{d x} [- 7 x])$

Step 16

Multiply $2$ by $6$ .

$\frac{2}{{(6 x^{2} - 7 x)}^{\frac{1}{2}}} (12 x + \frac{d}{d x} [- 7 x])$

Step 17

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x$ with respect to $x$ is $- 7 \frac{d}{d x} [x]$ .

$\frac{2}{{(6 x^{2} - 7 x)}^{\frac{1}{2}}} (12 x - 7 \frac{d}{d x} [x])$

Step 18

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

$\frac{2}{{(6 x^{2} - 7 x)}^{\frac{1}{2}}} (12 x - 7 \cdot 1)$

Step 19

Multiply $- 7$ by $1$ .

$\frac{2}{{(6 x^{2} - 7 x)}^{\frac{1}{2}}} (12 x - 7)$

Step 20

Simplify.

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Step 20.1

Reorder the factors of $\frac{2}{{(6 x^{2} - 7 x)}^{\frac{1}{2}}} (12 x - 7)$ .

$(12 x - 7) \frac{2}{{(6 x^{2} - 7 x)}^{\frac{1}{2}}}$

Step 20.2

Multiply $12 x - 7$ by $\frac{2}{{(6 x^{2} - 7 x)}^{\frac{1}{2}}}$ .

$\frac{(12 x - 7) \cdot 2}{{(6 x^{2} - 7 x)}^{\frac{1}{2}}}$

Step 20.3

Move $2$ to the left of $12 x - 7$ .

$\frac{2 (12 x - 7)}{{(6 x^{2} - 7 x)}^{\frac{1}{2}}}$