Calculus Examples

$y = \sqrt{3 x + 5} {(8 x - 3)}^{4}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x + 5}$ as ${(3 x + 5)}^{\frac{1}{2}}$ .

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

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 $f (x) = {(3 x + 5)}^{\frac{1}{2}}$ and $g (x) = {(8 x - 3)}^{4}$ .

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

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = 8 x - 3$ .

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Step 3.1

To apply the Chain Rule, set $u_{1}$ as $8 x - 3$ .

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

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 4$ .

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

Step 3.3

Replace all occurrences of $u_{1}$ with $8 x - 3$ .

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

Step 4

Differentiate.

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Step 4.1

Move $4$ to the left of ${(3 x + 5)}^{\frac{1}{2}}$ .

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Multiply $8$ by $1$ .

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

Step 4.6

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

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

Step 4.7

Simplify the expression.

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Step 4.7.1

Add $8$ and $0$ .

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

Step 4.7.2

Multiply $8$ by $4$ .

$32 {(3 x + 5)}^{\frac{1}{2}} {(8 x - 3)}^{3} + {(8 x - 3)}^{4} \frac{d}{d x} [{(3 x + 5)}^{\frac{1}{2}}]$

Step 5

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 $g (x) = 3 x + 5$ .

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Step 5.1

To apply the Chain Rule, set $u_{2}$ as $3 x + 5$ .

$32 {(3 x + 5)}^{\frac{1}{2}} {(8 x - 3)}^{3} + {(8 x - 3)}^{4} (\frac{d}{d u_{2}} [{u_{2}}^{\frac{1}{2}}] \frac{d}{d x} [3 x + 5])$

Step 5.2

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

$32 {(3 x + 5)}^{\frac{1}{2}} {(8 x - 3)}^{3} + {(8 x - 3)}^{4} (\frac{1}{2} {u_{2}}^{\frac{1}{2} - 1} \frac{d}{d x} [3 x + 5])$

Step 5.3

Replace all occurrences of $u_{2}$ with $3 x + 5$ .

$32 {(3 x + 5)}^{\frac{1}{2}} {(8 x - 3)}^{3} + {(8 x - 3)}^{4} (\frac{1}{2} {(3 x + 5)}^{\frac{1}{2} - 1} \frac{d}{d x} [3 x + 5])$

Step 6

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

$32 {(3 x + 5)}^{\frac{1}{2}} {(8 x - 3)}^{3} + {(8 x - 3)}^{4} (\frac{1}{2} {(3 x + 5)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [3 x + 5])$

Step 7

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

$32 {(3 x + 5)}^{\frac{1}{2}} {(8 x - 3)}^{3} + {(8 x - 3)}^{4} (\frac{1}{2} {(3 x + 5)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [3 x + 5])$

Step 8

Combine the numerators over the common denominator.

$32 {(3 x + 5)}^{\frac{1}{2}} {(8 x - 3)}^{3} + {(8 x - 3)}^{4} (\frac{1}{2} {(3 x + 5)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [3 x + 5])$

Step 9

Simplify the numerator.

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Step 9.1

Multiply $- 1$ by $2$ .

$32 {(3 x + 5)}^{\frac{1}{2}} {(8 x - 3)}^{3} + {(8 x - 3)}^{4} (\frac{1}{2} {(3 x + 5)}^{\frac{1 - 2}{2}} \frac{d}{d x} [3 x + 5])$

Step 9.2

Subtract $2$ from $1$ .

$32 {(3 x + 5)}^{\frac{1}{2}} {(8 x - 3)}^{3} + {(8 x - 3)}^{4} (\frac{1}{2} {(3 x + 5)}^{\frac{- 1}{2}} \frac{d}{d x} [3 x + 5])$

Step 10

Combine fractions.

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Step 10.1

Move the negative in front of the fraction.

$32 {(3 x + 5)}^{\frac{1}{2}} {(8 x - 3)}^{3} + {(8 x - 3)}^{4} (\frac{1}{2} {(3 x + 5)}^{- \frac{1}{2}} \frac{d}{d x} [3 x + 5])$

Step 10.2

Combine $\frac{1}{2}$ and ${(3 x + 5)}^{- \frac{1}{2}}$ .

$32 {(3 x + 5)}^{\frac{1}{2}} {(8 x - 3)}^{3} + {(8 x - 3)}^{4} (\frac{{(3 x + 5)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [3 x + 5])$

Step 10.3

Move ${(3 x + 5)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$32 {(3 x + 5)}^{\frac{1}{2}} {(8 x - 3)}^{3} + {(8 x - 3)}^{4} (\frac{1}{2 {(3 x + 5)}^{\frac{1}{2}}} \frac{d}{d x} [3 x + 5])$

Step 10.4

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

$32 {(3 x + 5)}^{\frac{1}{2}} {(8 x - 3)}^{3} + \frac{{(8 x - 3)}^{4}}{2 {(3 x + 5)}^{\frac{1}{2}}} \frac{d}{d x} [3 x + 5]$

Step 11

By the Sum Rule, the derivative of $3 x + 5$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [5]$ .

$32 {(3 x + 5)}^{\frac{1}{2}} {(8 x - 3)}^{3} + \frac{{(8 x - 3)}^{4}}{2 {(3 x + 5)}^{\frac{1}{2}}} (\frac{d}{d x} [3 x] + \frac{d}{d x} [5])$

Step 12

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

$32 {(3 x + 5)}^{\frac{1}{2}} {(8 x - 3)}^{3} + \frac{{(8 x - 3)}^{4}}{2 {(3 x + 5)}^{\frac{1}{2}}} (3 \frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 13

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

$32 {(3 x + 5)}^{\frac{1}{2}} {(8 x - 3)}^{3} + \frac{{(8 x - 3)}^{4}}{2 {(3 x + 5)}^{\frac{1}{2}}} (3 \cdot 1 + \frac{d}{d x} [5])$

Step 14

Multiply $3$ by $1$ .

$32 {(3 x + 5)}^{\frac{1}{2}} {(8 x - 3)}^{3} + \frac{{(8 x - 3)}^{4}}{2 {(3 x + 5)}^{\frac{1}{2}}} (3 + \frac{d}{d x} [5])$

Step 15

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

$32 {(3 x + 5)}^{\frac{1}{2}} {(8 x - 3)}^{3} + \frac{{(8 x - 3)}^{4}}{2 {(3 x + 5)}^{\frac{1}{2}}} (3 + 0)$

Step 16

Combine fractions.

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Step 16.1

Add $3$ and $0$ .

$32 {(3 x + 5)}^{\frac{1}{2}} {(8 x - 3)}^{3} + \frac{{(8 x - 3)}^{4}}{2 {(3 x + 5)}^{\frac{1}{2}}} \cdot 3$

Step 16.2

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

$32 {(3 x + 5)}^{\frac{1}{2}} {(8 x - 3)}^{3} + \frac{{(8 x - 3)}^{4} \cdot 3}{2 {(3 x + 5)}^{\frac{1}{2}}}$

Step 16.3

Move $3$ to the left of ${(8 x - 3)}^{4}$ .

$32 {(3 x + 5)}^{\frac{1}{2}} {(8 x - 3)}^{3} + \frac{3 \cdot {(8 x - 3)}^{4}}{2 {(3 x + 5)}^{\frac{1}{2}}}$

Step 17

To write $32 {(3 x + 5)}^{\frac{1}{2}} {(8 x - 3)}^{3}$ as a fraction with a common denominator, multiply by $\frac{2 {(3 x + 5)}^{\frac{1}{2}}}{2 {(3 x + 5)}^{\frac{1}{2}}}$ .

$32 {(3 x + 5)}^{\frac{1}{2}} {(8 x - 3)}^{3} \cdot \frac{2 {(3 x + 5)}^{\frac{1}{2}}}{2 {(3 x + 5)}^{\frac{1}{2}}} + \frac{3 {(8 x - 3)}^{4}}{2 {(3 x + 5)}^{\frac{1}{2}}}$

Step 18

Combine $32 {(3 x + 5)}^{\frac{1}{2}} {(8 x - 3)}^{3}$ and $\frac{2 {(3 x + 5)}^{\frac{1}{2}}}{2 {(3 x + 5)}^{\frac{1}{2}}}$ .

$\frac{32 {(3 x + 5)}^{\frac{1}{2}} {(8 x - 3)}^{3} (2 {(3 x + 5)}^{\frac{1}{2}})}{2 {(3 x + 5)}^{\frac{1}{2}}} + \frac{3 {(8 x - 3)}^{4}}{2 {(3 x + 5)}^{\frac{1}{2}}}$

Step 19

Combine the numerators over the common denominator.

$\frac{32 {(3 x + 5)}^{\frac{1}{2}} {(8 x - 3)}^{3} (2 {(3 x + 5)}^{\frac{1}{2}}) + 3 {(8 x - 3)}^{4}}{2 {(3 x + 5)}^{\frac{1}{2}}}$

Step 20

Multiply $2$ by $32$ .

$\frac{64 {(3 x + 5)}^{\frac{1}{2}} {(8 x - 3)}^{3} {(3 x + 5)}^{\frac{1}{2}} + 3 {(8 x - 3)}^{4}}{2 {(3 x + 5)}^{\frac{1}{2}}}$

Step 21

Multiply ${(3 x + 5)}^{\frac{1}{2}}$ by ${(3 x + 5)}^{\frac{1}{2}}$ by adding the exponents.

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Step 21.1

Move ${(3 x + 5)}^{\frac{1}{2}}$ .

$\frac{64 ({(3 x + 5)}^{\frac{1}{2}} {(3 x + 5)}^{\frac{1}{2}}) {(8 x - 3)}^{3} + 3 {(8 x - 3)}^{4}}{2 {(3 x + 5)}^{\frac{1}{2}}}$

Step 21.2

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

$\frac{64 {(3 x + 5)}^{\frac{1}{2} + \frac{1}{2}} {(8 x - 3)}^{3} + 3 {(8 x - 3)}^{4}}{2 {(3 x + 5)}^{\frac{1}{2}}}$

Step 21.3

Combine the numerators over the common denominator.

$\frac{64 {(3 x + 5)}^{\frac{1 + 1}{2}} {(8 x - 3)}^{3} + 3 {(8 x - 3)}^{4}}{2 {(3 x + 5)}^{\frac{1}{2}}}$

Step 21.4

Add $1$ and $1$ .

$\frac{64 {(3 x + 5)}^{\frac{2}{2}} {(8 x - 3)}^{3} + 3 {(8 x - 3)}^{4}}{2 {(3 x + 5)}^{\frac{1}{2}}}$

Step 21.5

Divide $2$ by $2$ .

$\frac{64 {(3 x + 5)}^{1} {(8 x - 3)}^{3} + 3 {(8 x - 3)}^{4}}{2 {(3 x + 5)}^{\frac{1}{2}}}$

Step 22

Simplify $64 {(3 x + 5)}^{1} {(8 x - 3)}^{3}$ .

$\frac{64 (3 x + 5) {(8 x - 3)}^{3} + 3 {(8 x - 3)}^{4}}{2 {(3 x + 5)}^{\frac{1}{2}}}$

Step 23

Simplify.

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Step 23.1

Apply the distributive property.

$\frac{(64 (3 x) + 64 \cdot 5) {(8 x - 3)}^{3} + 3 {(8 x - 3)}^{4}}{2 {(3 x + 5)}^{\frac{1}{2}}}$

Step 23.2

Simplify the numerator.

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Step 23.2.1

Factor ${(8 x - 3)}^{3}$ out of $(64 \cdot 3 x + 64 \cdot 5) {(8 x - 3)}^{3} + 3 {(8 x - 3)}^{4}$ .

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Step 23.2.1.1

Factor ${(8 x - 3)}^{3}$ out of $(64 \cdot 3 x + 64 \cdot 5) {(8 x - 3)}^{3}$ .

$\frac{{(8 x - 3)}^{3} (64 \cdot 3 x + 64 \cdot 5) + 3 {(8 x - 3)}^{4}}{2 {(3 x + 5)}^{\frac{1}{2}}}$

Step 23.2.1.2

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

$\frac{{(8 x - 3)}^{3} (64 \cdot 3 x + 64 \cdot 5) + {(8 x - 3)}^{3} (3 (8 x - 3))}{2 {(3 x + 5)}^{\frac{1}{2}}}$

Step 23.2.1.3

Factor ${(8 x - 3)}^{3}$ out of ${(8 x - 3)}^{3} (64 \cdot 3 x + 64 \cdot 5) + {(8 x - 3)}^{3} (3 (8 x - 3))$ .

$\frac{{(8 x - 3)}^{3} (64 \cdot 3 x + 64 \cdot 5 + 3 (8 x - 3))}{2 {(3 x + 5)}^{\frac{1}{2}}}$

Step 23.2.2

Multiply $64$ by $3$ .

$\frac{{(8 x - 3)}^{3} (192 x + 64 \cdot 5 + 3 (8 x - 3))}{2 {(3 x + 5)}^{\frac{1}{2}}}$

Step 23.2.3

Multiply $64$ by $5$ .

$\frac{{(8 x - 3)}^{3} (192 x + 320 + 3 (8 x - 3))}{2 {(3 x + 5)}^{\frac{1}{2}}}$

Step 23.2.4

Apply the distributive property.

$\frac{{(8 x - 3)}^{3} (192 x + 320 + 3 (8 x) + 3 \cdot - 3)}{2 {(3 x + 5)}^{\frac{1}{2}}}$

Step 23.2.5

Multiply $8$ by $3$ .

$\frac{{(8 x - 3)}^{3} (192 x + 320 + 24 x + 3 \cdot - 3)}{2 {(3 x + 5)}^{\frac{1}{2}}}$

Step 23.2.6

Multiply $3$ by $- 3$ .

$\frac{{(8 x - 3)}^{3} (192 x + 320 + 24 x - 9)}{2 {(3 x + 5)}^{\frac{1}{2}}}$

Step 23.2.7

Add $192 x$ and $24 x$ .

$\frac{{(8 x - 3)}^{3} (216 x + 320 - 9)}{2 {(3 x + 5)}^{\frac{1}{2}}}$

Step 23.2.8

Subtract $9$ from $320$ .

$\frac{{(8 x - 3)}^{3} (216 x + 311)}{2 {(3 x + 5)}^{\frac{1}{2}}}$