Calculus Examples

$f (x) = 4 x^{2} \sqrt{3 x - 8}$

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

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

Step 1.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2} {(3 x - 8)}^{\frac{1}{2}}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2} {(3 x - 8)}^{\frac{1}{2}}]$ .

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

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

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

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

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

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

Step 9

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

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

Step 10

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

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

Step 11

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

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

Step 12

Multiply $3$ by $1$ .

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

Step 13

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8$ with respect to $x$ is $0$ .

$4 (\frac{x^{2}}{2 {(3 x - 8)}^{\frac{1}{2}}} (3 + 0) + {(3 x - 8)}^{\frac{1}{2}} \frac{d}{d x} [x^{2}])$

Step 14

Combine fractions.

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Step 14.1

Add $3$ and $0$ .

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

Step 14.2

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

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

Step 14.3

Move $3$ to the left of $x^{2}$ .

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

Step 15

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

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

Step 16

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

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

Step 17

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

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Step 17.1

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

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

Step 17.2

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

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

Step 17.3

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

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

Step 17.4

Combine the numerators over the common denominator.

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

Step 18

Multiply $2$ by $2$ .

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

Step 19

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

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Step 19.1

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

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

Step 19.2

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

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

Step 19.3

Combine the numerators over the common denominator.

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

Step 19.4

Add $1$ and $1$ .

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

Step 19.5

Divide $2$ by $2$ .

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

Step 20

Simplify $4 x {(3 x - 8)}^{1}$ .

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

Step 21

Combine $4$ and $\frac{3 x^{2} + 4 x (3 x - 8)}{2 {(3 x - 8)}^{\frac{1}{2}}}$ .

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

Step 22

Factor $2$ out of $4 (3 x^{2} + 4 x (3 x - 8))$ .

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

Step 23

Cancel the common factors.

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Step 23.1

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

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

Step 23.2

Cancel the common factor.

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

Step 23.3

Rewrite the expression.

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

Step 24

Simplify.

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Step 24.1

Apply the distributive property.

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

Step 24.2

Apply the distributive property.

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

Step 24.3

Simplify the numerator.

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Step 24.3.1

Simplify each term.

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Step 24.3.1.1

Multiply $3$ by $2$ .

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

Step 24.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 24.3.1.2.1

Move $x$ .

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

Step 24.3.1.2.2

Multiply $x$ by $x$ .

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

Step 24.3.1.3

Multiply $3$ by $4$ .

$\frac{6 x^{2} + 2 (12 x^{2}) + 2 (4 x \cdot - 8)}{{(3 x - 8)}^{\frac{1}{2}}}$

Step 24.3.1.4

Multiply $12$ by $2$ .

$\frac{6 x^{2} + 24 x^{2} + 2 (4 x \cdot - 8)}{{(3 x - 8)}^{\frac{1}{2}}}$

Step 24.3.1.5

Multiply $- 8$ by $4$ .

$\frac{6 x^{2} + 24 x^{2} + 2 (- 32 x)}{{(3 x - 8)}^{\frac{1}{2}}}$

Step 24.3.1.6

Multiply $- 32$ by $2$ .

$\frac{6 x^{2} + 24 x^{2} - 64 x}{{(3 x - 8)}^{\frac{1}{2}}}$

Step 24.3.2

Add $6 x^{2}$ and $24 x^{2}$ .

$\frac{30 x^{2} - 64 x}{{(3 x - 8)}^{\frac{1}{2}}}$

Step 24.4

Factor $2 x$ out of $30 x^{2} - 64 x$ .

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Step 24.4.1

Factor $2 x$ out of $30 x^{2}$ .

$\frac{2 x (15 x) - 64 x}{{(3 x - 8)}^{\frac{1}{2}}}$

Step 24.4.2

Factor $2 x$ out of $- 64 x$ .

$\frac{2 x (15 x) + 2 x (- 32)}{{(3 x - 8)}^{\frac{1}{2}}}$

Step 24.4.3

Factor $2 x$ out of $2 x (15 x) + 2 x (- 32)$ .

$\frac{2 x (15 x - 32)}{{(3 x - 8)}^{\frac{1}{2}}}$