Calculus Examples

$f (x) = 6 \sqrt{8 x^{2} + 5 x}$

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 $\sqrt{8 x^{2} + 5 x}$ as ${(8 x^{2} + 5 x)}^{\frac{1}{2}}$ .

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

Step 1.2

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

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

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $1$ .

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

Step 7

Move the negative in front of the fraction.

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Factor $2$ out of $6$ .

$\frac{2 \cdot 3}{2 {(8 x^{2} + 5 x)}^{\frac{1}{2}}} \frac{d}{d x} [8 x^{2} + 5 x]$

Step 12

Cancel the common factors.

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Step 12.1

Factor $2$ out of $2 {(8 x^{2} + 5 x)}^{\frac{1}{2}}$ .

$\frac{2 \cdot 3}{2 ({(8 x^{2} + 5 x)}^{\frac{1}{2}})} \frac{d}{d x} [8 x^{2} + 5 x]$

Step 12.2

Cancel the common factor.

$\frac{2 \cdot 3}{2 {(8 x^{2} + 5 x)}^{\frac{1}{2}}} \frac{d}{d x} [8 x^{2} + 5 x]$

Step 12.3

Rewrite the expression.

$\frac{3}{{(8 x^{2} + 5 x)}^{\frac{1}{2}}} \frac{d}{d x} [8 x^{2} + 5 x]$

Step 13

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

$\frac{3}{{(8 x^{2} + 5 x)}^{\frac{1}{2}}} (\frac{d}{d x} [8 x^{2}] + \frac{d}{d x} [5 x])$

Step 14

Since $8$ is constant with respect to $x$ , the derivative of $8 x^{2}$ with respect to $x$ is $8 \frac{d}{d x} [x^{2}]$ .

$\frac{3}{{(8 x^{2} + 5 x)}^{\frac{1}{2}}} (8 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [5 x])$

Step 15

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

$\frac{3}{{(8 x^{2} + 5 x)}^{\frac{1}{2}}} (8 (2 x) + \frac{d}{d x} [5 x])$

Step 16

Multiply $2$ by $8$ .

$\frac{3}{{(8 x^{2} + 5 x)}^{\frac{1}{2}}} (16 x + \frac{d}{d x} [5 x])$

Step 17

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$\frac{3}{{(8 x^{2} + 5 x)}^{\frac{1}{2}}} (16 x + 5 \frac{d}{d x} [x])$

Step 18

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

$\frac{3}{{(8 x^{2} + 5 x)}^{\frac{1}{2}}} (16 x + 5 \cdot 1)$

Step 19

Multiply $5$ by $1$ .

$\frac{3}{{(8 x^{2} + 5 x)}^{\frac{1}{2}}} (16 x + 5)$

Step 20

Simplify.

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Step 20.1

Reorder the factors of $\frac{3}{{(8 x^{2} + 5 x)}^{\frac{1}{2}}} (16 x + 5)$ .

$(16 x + 5) \frac{3}{{(8 x^{2} + 5 x)}^{\frac{1}{2}}}$

Step 20.2

Multiply $16 x + 5$ by $\frac{3}{{(8 x^{2} + 5 x)}^{\frac{1}{2}}}$ .

$\frac{(16 x + 5) \cdot 3}{{(8 x^{2} + 5 x)}^{\frac{1}{2}}}$

Step 20.3

Move $3$ to the left of $16 x + 5$ .

$\frac{3 (16 x + 5)}{{(8 x^{2} + 5 x)}^{\frac{1}{2}}}$