Calculus Examples

$y = 2 \sqrt{6 x^{2} - 8 x + 5}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 x^{2} - 8 x + 5}$ as ${(6 x^{2} - 8 x + 5)}^{\frac{1}{2}}$ .

$y = 2 {(6 x^{2} - 8 x + 5)}^{\frac{1}{2}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (2 {(6 x^{2} - 8 x + 5)}^{\frac{1}{2}})$

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Since $2$ is constant with respect to $x$ , the derivative of $2 {(6 x^{2} - 8 x + 5)}^{\frac{1}{2}}$ with respect to $x$ is $2 \frac{d}{d x} [{(6 x^{2} - 8 x + 5)}^{\frac{1}{2}}]$ .

$2 \frac{d}{d x} [{(6 x^{2} - 8 x + 5)}^{\frac{1}{2}}]$

Step 4.2

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 $g (x) = 6 x^{2} - 8 x + 5$ .

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Step 4.2.1

To apply the Chain Rule, set $u$ as $6 x^{2} - 8 x + 5$ .

$2 (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [6 x^{2} - 8 x + 5])$

Step 4.2.2

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

$2 (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [6 x^{2} - 8 x + 5])$

Step 4.2.3

Replace all occurrences of $u$ with $6 x^{2} - 8 x + 5$ .

$2 (\frac{1}{2} {(6 x^{2} - 8 x + 5)}^{\frac{1}{2} - 1} \frac{d}{d x} [6 x^{2} - 8 x + 5])$

Step 4.3

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

$2 (\frac{1}{2} {(6 x^{2} - 8 x + 5)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [6 x^{2} - 8 x + 5])$

Step 4.4

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

$2 (\frac{1}{2} {(6 x^{2} - 8 x + 5)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [6 x^{2} - 8 x + 5])$

Step 4.5

Combine the numerators over the common denominator.

$2 (\frac{1}{2} {(6 x^{2} - 8 x + 5)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [6 x^{2} - 8 x + 5])$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 1$ by $2$ .

$2 (\frac{1}{2} {(6 x^{2} - 8 x + 5)}^{\frac{1 - 2}{2}} \frac{d}{d x} [6 x^{2} - 8 x + 5])$

Step 4.6.2

Subtract $2$ from $1$ .

$2 (\frac{1}{2} {(6 x^{2} - 8 x + 5)}^{\frac{- 1}{2}} \frac{d}{d x} [6 x^{2} - 8 x + 5])$

Step 4.7

Simplify terms.

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Step 4.7.1

Move the negative in front of the fraction.

$2 (\frac{1}{2} {(6 x^{2} - 8 x + 5)}^{- \frac{1}{2}} \frac{d}{d x} [6 x^{2} - 8 x + 5])$

Step 4.7.2

Combine $\frac{1}{2}$ and ${(6 x^{2} - 8 x + 5)}^{- \frac{1}{2}}$ .

$2 (\frac{{(6 x^{2} - 8 x + 5)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [6 x^{2} - 8 x + 5])$

Step 4.7.3

Move ${(6 x^{2} - 8 x + 5)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 (\frac{1}{2 {(6 x^{2} - 8 x + 5)}^{\frac{1}{2}}} \frac{d}{d x} [6 x^{2} - 8 x + 5])$

Step 4.7.4

Combine $\frac{1}{2 {(6 x^{2} - 8 x + 5)}^{\frac{1}{2}}}$ and $2$ .

$\frac{2}{2 {(6 x^{2} - 8 x + 5)}^{\frac{1}{2}}} \frac{d}{d x} [6 x^{2} - 8 x + 5]$

Step 4.7.5

Cancel the common factor.

$\frac{2}{2 {(6 x^{2} - 8 x + 5)}^{\frac{1}{2}}} \frac{d}{d x} [6 x^{2} - 8 x + 5]$

Step 4.7.6

Rewrite the expression.

$\frac{1}{{(6 x^{2} - 8 x + 5)}^{\frac{1}{2}}} \frac{d}{d x} [6 x^{2} - 8 x + 5]$

Step 4.8

By the Sum Rule, the derivative of $6 x^{2} - 8 x + 5$ with respect to $x$ is $\frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [5]$ .

$\frac{1}{{(6 x^{2} - 8 x + 5)}^{\frac{1}{2}}} (\frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [5])$

Step 4.9

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

$\frac{1}{{(6 x^{2} - 8 x + 5)}^{\frac{1}{2}}} (6 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [5])$

Step 4.10

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

$\frac{1}{{(6 x^{2} - 8 x + 5)}^{\frac{1}{2}}} (6 (2 x) + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [5])$

Step 4.11

Multiply $2$ by $6$ .

$\frac{1}{{(6 x^{2} - 8 x + 5)}^{\frac{1}{2}}} (12 x + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [5])$

Step 4.12

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x$ with respect to $x$ is $- 8 \frac{d}{d x} [x]$ .

$\frac{1}{{(6 x^{2} - 8 x + 5)}^{\frac{1}{2}}} (12 x - 8 \frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 4.13

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

$\frac{1}{{(6 x^{2} - 8 x + 5)}^{\frac{1}{2}}} (12 x - 8 \cdot 1 + \frac{d}{d x} [5])$

Step 4.14

Multiply $- 8$ by $1$ .

$\frac{1}{{(6 x^{2} - 8 x + 5)}^{\frac{1}{2}}} (12 x - 8 + \frac{d}{d x} [5])$

Step 4.15

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$\frac{1}{{(6 x^{2} - 8 x + 5)}^{\frac{1}{2}}} (12 x - 8 + 0)$

Step 4.16

Add $12 x - 8$ and $0$ .

$\frac{1}{{(6 x^{2} - 8 x + 5)}^{\frac{1}{2}}} (12 x - 8)$

Step 4.17

Simplify.

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Step 4.17.1

Reorder the factors of $\frac{1}{{(6 x^{2} - 8 x + 5)}^{\frac{1}{2}}} (12 x - 8)$ .

$(12 x - 8) \frac{1}{{(6 x^{2} - 8 x + 5)}^{\frac{1}{2}}}$

Step 4.17.2

Multiply $12 x - 8$ by $\frac{1}{{(6 x^{2} - 8 x + 5)}^{\frac{1}{2}}}$ .

$\frac{12 x - 8}{{(6 x^{2} - 8 x + 5)}^{\frac{1}{2}}}$

Step 4.17.3

Factor $4$ out of $12 x - 8$ .

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Step 4.17.3.1

Factor $4$ out of $12 x$ .

$\frac{4 (3 x) - 8}{{(6 x^{2} - 8 x + 5)}^{\frac{1}{2}}}$

Step 4.17.3.2

Factor $4$ out of $- 8$ .

$\frac{4 (3 x) + 4 (- 2)}{{(6 x^{2} - 8 x + 5)}^{\frac{1}{2}}}$

Step 4.17.3.3

Factor $4$ out of $4 (3 x) + 4 (- 2)$ .

$\frac{4 (3 x - 2)}{{(6 x^{2} - 8 x + 5)}^{\frac{1}{2}}}$

Step 5

Reform the equation by setting the left side equal to the right side.

$y' = \frac{4 (3 x - 2)}{{(6 x^{2} - 8 x + 5)}^{\frac{1}{2}}}$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{4 (3 x - 2)}{{(6 x^{2} - 8 x + 5)}^{\frac{1}{2}}}$