Calculus Examples

$u = v^{2} \sqrt{8 v - 3}$

Step 1

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

$u = {u'}^{2} {(8 u' - 3)}^{\frac{1}{2}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d u'} u = \frac{d}{d u'} ({u'}^{2} {(8 u' - 3)}^{\frac{1}{2}})$

Step 3

The derivative of $u$ with respect to $u'$ is $u'$ .

$u'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

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

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

Step 4.2

Differentiate using the chain rule, which states that $\frac{d}{d v} [f (g (u'))]$ is $f' (g (u')) g' (u')$ where $f (u') = {u'}^{\frac{1}{2}}$ and $g (u') = 8 u' - 3$ .

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.2.3

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

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

Step 4.3

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

${u'}^{2} (\frac{1}{2} {(8 u' - 3)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d v} [8 u' - 3]) + {(8 u' - 3)}^{\frac{1}{2}} \frac{d}{d v} [{u'}^{2}]$

Step 4.4

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

${u'}^{2} (\frac{1}{2} {(8 u' - 3)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d v} [8 u' - 3]) + {(8 u' - 3)}^{\frac{1}{2}} \frac{d}{d v} [{u'}^{2}]$

Step 4.5

Combine the numerators over the common denominator.

${u'}^{2} (\frac{1}{2} {(8 u' - 3)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d v} [8 u' - 3]) + {(8 u' - 3)}^{\frac{1}{2}} \frac{d}{d v} [{u'}^{2}]$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 1$ by $2$ .

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

Step 4.6.2

Subtract $2$ from $1$ .

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

Step 4.7

Combine fractions.

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Step 4.7.1

Move the negative in front of the fraction.

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

Step 4.7.2

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

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

Step 4.7.3

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

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

Step 4.7.4

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

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

Step 4.8

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

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

Step 4.9

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

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

Step 4.10

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

$\frac{{u'}^{2}}{2 {(8 u' - 3)}^{\frac{1}{2}}} (8 \cdot 1 + \frac{d}{d v} [- 3]) + {(8 u' - 3)}^{\frac{1}{2}} \frac{d}{d v} [{u'}^{2}]$

Step 4.11

Multiply $8$ by $1$ .

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

Step 4.12

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

$\frac{{u'}^{2}}{2 {(8 u' - 3)}^{\frac{1}{2}}} (8 + 0) + {(8 u' - 3)}^{\frac{1}{2}} \frac{d}{d v} [{u'}^{2}]$

Step 4.13

Simplify terms.

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Step 4.13.1

Add $8$ and $0$ .

$\frac{{u'}^{2}}{2 {(8 u' - 3)}^{\frac{1}{2}}} \cdot 8 + {(8 u' - 3)}^{\frac{1}{2}} \frac{d}{d v} [{u'}^{2}]$

Step 4.13.2

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

$\frac{{u'}^{2} \cdot 8}{2 {(8 u' - 3)}^{\frac{1}{2}}} + {(8 u' - 3)}^{\frac{1}{2}} \frac{d}{d v} [{u'}^{2}]$

Step 4.13.3

Move $8$ to the left of ${u'}^{2}$ .

$\frac{8 \cdot {u'}^{2}}{2 {(8 u' - 3)}^{\frac{1}{2}}} + {(8 u' - 3)}^{\frac{1}{2}} \frac{d}{d v} [{u'}^{2}]$

Step 4.13.4

Factor $2$ out of $8 {u'}^{2}$ .

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

Step 4.14

Cancel the common factors.

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Step 4.14.1

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

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

Step 4.14.2

Cancel the common factor.

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

Step 4.14.3

Rewrite the expression.

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

Step 4.15

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

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

Step 4.16

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

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

Step 4.17

Combine $\frac{4 {u'}^{2}}{{(8 u' - 3)}^{\frac{1}{2}}}$ and $2 {(8 u' - 3)}^{\frac{1}{2}} u'$ using a common denominator.

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Step 4.17.1

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

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

Step 4.17.2

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

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

Step 4.17.3

Combine the numerators over the common denominator.

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

Step 4.18

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

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Step 4.18.1

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

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

Step 4.18.2

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

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

Step 4.18.3

Combine the numerators over the common denominator.

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

Step 4.18.4

Add $1$ and $1$ .

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

Step 4.18.5

Divide $2$ by $2$ .

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

Step 4.19

Simplify $2 u' {(8 u' - 3)}^{1}$ .

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

Step 4.20

Simplify.

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Step 4.20.1

Apply the distributive property.

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

Step 4.20.2

Simplify the numerator.

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Step 4.20.2.1

Simplify each term.

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Step 4.20.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 4.20.2.1.2

Multiply $u'$ by $u'$ by adding the exponents.

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Step 4.20.2.1.2.1

Move $u'$ .

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

Step 4.20.2.1.2.2

Multiply $u'$ by $u'$ .

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

Step 4.20.2.1.3

Multiply $2$ by $8$ .

$\frac{4 {u'}^{2} + 16 {u'}^{2} + 2 u' \cdot - 3}{{(8 u' - 3)}^{\frac{1}{2}}}$

Step 4.20.2.1.4

Multiply $- 3$ by $2$ .

$\frac{4 {u'}^{2} + 16 {u'}^{2} - 6 u'}{{(8 u' - 3)}^{\frac{1}{2}}}$

Step 4.20.2.2

Add $4 {u'}^{2}$ and $16 {u'}^{2}$ .

$\frac{20 {u'}^{2} - 6 u'}{{(8 u' - 3)}^{\frac{1}{2}}}$

Step 4.20.3

Factor $2 u'$ out of $20 {u'}^{2} - 6 u'$ .

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Step 4.20.3.1

Factor $2 u'$ out of $20 {u'}^{2}$ .

$\frac{2 u' (10 u') - 6 u'}{{(8 u' - 3)}^{\frac{1}{2}}}$

Step 4.20.3.2

Factor $2 u'$ out of $- 6 u'$ .

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

Step 4.20.3.3

Factor $2 u'$ out of $2 u' (10 u') + 2 u' \cdot - 3$ .

$\frac{2 u' (10 u' - 3)}{{(8 u' - 3)}^{\frac{1}{2}}}$

Step 5

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

$u' = \frac{2 u' (10 u' - 3)}{{(8 u' - 3)}^{\frac{1}{2}}}$

Step 6

Graph each side of the equation. The solution is the x-value of the point of intersection.

No solution

Step 7

Replace $u'$ with $\frac{d u}{d v}$ .

$\frac{d u}{d v}$