Calculus Examples

$y = (6 x - 5) \sqrt{8 x - 3}$ ; $x = 1$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{8 x - 3}$ as ${(8 x - 3)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [(6 x - 5) {(8 x - 3)}^{\frac{1}{2}}]$

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = 6 x - 5$ and $g (x) = {(8 x - 3)}^{\frac{1}{2}}$ .

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

Step 3

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 $g (x) = 8 x - 3$ .

Tap for more steps...

Step 3.1

To apply the Chain Rule, set $u$ as $8 x - 3$ .

$(6 x - 5) (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [8 x - 3]) + {(8 x - 3)}^{\frac{1}{2}} \frac{d}{d x} [6 x - 5]$

Step 3.2

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

$(6 x - 5) (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [8 x - 3]) + {(8 x - 3)}^{\frac{1}{2}} \frac{d}{d x} [6 x - 5]$

Step 3.3

Replace all occurrences of $u$ with $8 x - 3$ .

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

Step 4

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

$(6 x - 5) (\frac{1}{2} {(8 x - 3)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [8 x - 3]) + {(8 x - 3)}^{\frac{1}{2}} \frac{d}{d x} [6 x - 5]$

Step 5

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

$(6 x - 5) (\frac{1}{2} {(8 x - 3)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [8 x - 3]) + {(8 x - 3)}^{\frac{1}{2}} \frac{d}{d x} [6 x - 5]$

Step 6

Combine the numerators over the common denominator.

$(6 x - 5) (\frac{1}{2} {(8 x - 3)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [8 x - 3]) + {(8 x - 3)}^{\frac{1}{2}} \frac{d}{d x} [6 x - 5]$

Step 7

Simplify the numerator.

Tap for more steps...

Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Combine fractions.

Tap for more steps...

Step 8.1

Move the negative in front of the fraction.

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

Step 8.2

Combine $\frac{1}{2}$ and ${(8 x - 3)}^{- \frac{1}{2}}$ .

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

Step 8.3

Move ${(8 x - 3)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 9

By the Sum Rule, the derivative of $8 x - 3$ with respect to $x$ is $\frac{d}{d x} [8 x] + \frac{d}{d x} [- 3]$ .

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

Step 10

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

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

Step 11

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

$(6 x - 5) \frac{1}{2 {(8 x - 3)}^{\frac{1}{2}}} (8 \cdot 1 + \frac{d}{d x} [- 3]) + {(8 x - 3)}^{\frac{1}{2}} \frac{d}{d x} [6 x - 5]$

Step 12

Multiply $8$ by $1$ .

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

Step 13

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$(6 x - 5) \frac{1}{2 {(8 x - 3)}^{\frac{1}{2}}} (8 + 0) + {(8 x - 3)}^{\frac{1}{2}} \frac{d}{d x} [6 x - 5]$

Step 14

Simplify terms.

Tap for more steps...

Step 14.1

Add $8$ and $0$ .

$(6 x - 5) \frac{1}{2 {(8 x - 3)}^{\frac{1}{2}}} \cdot 8 + {(8 x - 3)}^{\frac{1}{2}} \frac{d}{d x} [6 x - 5]$

Step 14.2

Combine $8$ and $\frac{1}{2 {(8 x - 3)}^{\frac{1}{2}}}$ .

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

Step 14.3

Factor $2$ out of $8$ .

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

Step 15

Cancel the common factors.

Tap for more steps...

Step 15.1

Factor $2$ out of $2 {(8 x - 3)}^{\frac{1}{2}}$ .

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

Step 15.2

Cancel the common factor.

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

Step 15.3

Rewrite the expression.

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

Step 16

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

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

Step 17

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

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

Step 18

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

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

Step 19

Multiply $6$ by $1$ .

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

Step 20

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

$(6 x - 5) \frac{4}{{(8 x - 3)}^{\frac{1}{2}}} + {(8 x - 3)}^{\frac{1}{2}} (6 + 0)$

Step 21

Simplify the expression.

Tap for more steps...

Step 21.1

Add $6$ and $0$ .

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

Step 21.2

Move $6$ to the left of ${(8 x - 3)}^{\frac{1}{2}}$ .

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

Step 22

Simplify.

Tap for more steps...

Step 22.1

Simplify each term.

Tap for more steps...

Step 22.1.1

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

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

Step 22.1.2

Move $4$ to the left of $6 x - 5$ .

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

Step 22.2

To write $6 {(8 x - 3)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(8 x - 3)}^{\frac{1}{2}}}{{(8 x - 3)}^{\frac{1}{2}}}$ .

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

Step 22.3

Combine the numerators over the common denominator.

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

Step 22.4

Simplify the numerator.

Tap for more steps...

Step 22.4.1

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

Tap for more steps...

Step 22.4.1.1

Factor $2$ out of $4 (6 x - 5)$ .

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

Step 22.4.1.2

Factor $2$ out of $6 {(8 x - 3)}^{\frac{1}{2}} {(8 x - 3)}^{\frac{1}{2}}$ .

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

Step 22.4.1.3

Factor $2$ out of $2 (2 (6 x - 5)) + 2 (3 {(8 x - 3)}^{\frac{1}{2}} {(8 x - 3)}^{\frac{1}{2}})$ .

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

Step 22.4.2

Apply the distributive property.

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

Step 22.4.3

Multiply $6$ by $2$ .

$\frac{2 (12 x + 2 \cdot - 5 + 3 {(8 x - 3)}^{\frac{1}{2}} {(8 x - 3)}^{\frac{1}{2}})}{{(8 x - 3)}^{\frac{1}{2}}}$

Step 22.4.4

Multiply $2$ by $- 5$ .

$\frac{2 (12 x - 10 + 3 {(8 x - 3)}^{\frac{1}{2}} {(8 x - 3)}^{\frac{1}{2}})}{{(8 x - 3)}^{\frac{1}{2}}}$

Step 22.4.5

Multiply ${(8 x - 3)}^{\frac{1}{2}}$ by ${(8 x - 3)}^{\frac{1}{2}}$ by adding the exponents.

Tap for more steps...

Step 22.4.5.1

Move ${(8 x - 3)}^{\frac{1}{2}}$ .

$\frac{2 (12 x - 10 + 3 ({(8 x - 3)}^{\frac{1}{2}} {(8 x - 3)}^{\frac{1}{2}}))}{{(8 x - 3)}^{\frac{1}{2}}}$

Step 22.4.5.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 (12 x - 10 + 3 {(8 x - 3)}^{\frac{1}{2} + \frac{1}{2}})}{{(8 x - 3)}^{\frac{1}{2}}}$

Step 22.4.5.3

Combine the numerators over the common denominator.

$\frac{2 (12 x - 10 + 3 {(8 x - 3)}^{\frac{1 + 1}{2}})}{{(8 x - 3)}^{\frac{1}{2}}}$

Step 22.4.5.4

Add $1$ and $1$ .

$\frac{2 (12 x - 10 + 3 {(8 x - 3)}^{\frac{2}{2}})}{{(8 x - 3)}^{\frac{1}{2}}}$

Step 22.4.5.5

Divide $2$ by $2$ .

$\frac{2 (12 x - 10 + 3 {(8 x - 3)}^{1})}{{(8 x - 3)}^{\frac{1}{2}}}$

Step 22.4.6

Simplify $3 {(8 x - 3)}^{1}$ .

$\frac{2 (12 x - 10 + 3 (8 x - 3))}{{(8 x - 3)}^{\frac{1}{2}}}$

Step 22.4.7

Apply the distributive property.

$\frac{2 (12 x - 10 + 3 (8 x) + 3 \cdot - 3)}{{(8 x - 3)}^{\frac{1}{2}}}$

Step 22.4.8

Multiply $8$ by $3$ .

$\frac{2 (12 x - 10 + 24 x + 3 \cdot - 3)}{{(8 x - 3)}^{\frac{1}{2}}}$

Step 22.4.9

Multiply $3$ by $- 3$ .

$\frac{2 (12 x - 10 + 24 x - 9)}{{(8 x - 3)}^{\frac{1}{2}}}$

Step 22.4.10

Add $12 x$ and $24 x$ .

$\frac{2 (36 x - 10 - 9)}{{(8 x - 3)}^{\frac{1}{2}}}$

Step 22.4.11

Subtract $9$ from $- 10$ .

$\frac{2 (36 x - 19)}{{(8 x - 3)}^{\frac{1}{2}}}$

Step 23

Evaluate the derivative at $x = 1$ .

$\frac{2 (36 (1) - 19)}{{(8 (1) - 3)}^{\frac{1}{2}}}$

Step 24

Simplify.

Tap for more steps...

Step 24.1

Simplify the numerator.

Tap for more steps...

Step 24.1.1

Multiply $36$ by $1$ .

$\frac{2 (36 - 19)}{{(8 (1) - 3)}^{\frac{1}{2}}}$

Step 24.1.2

Subtract $19$ from $36$ .

$\frac{2 \cdot 17}{{(8 (1) - 3)}^{\frac{1}{2}}}$

Step 24.2

Simplify the denominator.

Tap for more steps...

Step 24.2.1

Multiply $8$ by $1$ .

$\frac{2 \cdot 17}{{(8 - 3)}^{\frac{1}{2}}}$

Step 24.2.2

Subtract $3$ from $8$ .

$\frac{2 \cdot 17}{5^{\frac{1}{2}}}$

Step 24.3

Multiply $2$ by $17$ .

$\frac{34}{5^{\frac{1}{2}}}$