Calculus Examples

$u = v^{2} \sqrt{2 v - 9}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 u' - 9}$ as ${(2 u' - 9)}^{\frac{1}{2}}$ .

$u = {u'}^{2} {(2 u' - 9)}^{\frac{1}{2}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d u'} u = \frac{d}{d u'} ({u'}^{2} {(2 u' - 9)}^{\frac{1}{2}})$

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 $g (u') = {(2 u' - 9)}^{\frac{1}{2}}$ .

${u'}^{2} \frac{d}{d v} [{(2 u' - 9)}^{\frac{1}{2}}] + {(2 u' - 9)}^{\frac{1}{2}} \frac{d}{d v} [{u'}^{2}]$

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 $g (u') = 2 u' - 9$ .

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Step 4.2.1

To apply the Chain Rule, set $u$ as $2 u' - 9$ .

${u'}^{2} (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d v} [2 u' - 9]) + {(2 u' - 9)}^{\frac{1}{2}} \frac{d}{d v} [{u'}^{2}]$

Step 4.2.2

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

${u'}^{2} (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d v} [2 u' - 9]) + {(2 u' - 9)}^{\frac{1}{2}} \frac{d}{d v} [{u'}^{2}]$

Step 4.2.3

Replace all occurrences of $u$ with $2 u' - 9$ .

${u'}^{2} (\frac{1}{2} {(2 u' - 9)}^{\frac{1}{2} - 1} \frac{d}{d v} [2 u' - 9]) + {(2 u' - 9)}^{\frac{1}{2}} \frac{d}{d v} [{u'}^{2}]$

Step 4.3

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

${u'}^{2} (\frac{1}{2} {(2 u' - 9)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d v} [2 u' - 9]) + {(2 u' - 9)}^{\frac{1}{2}} \frac{d}{d v} [{u'}^{2}]$

Step 4.4

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

${u'}^{2} (\frac{1}{2} {(2 u' - 9)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d v} [2 u' - 9]) + {(2 u' - 9)}^{\frac{1}{2}} \frac{d}{d v} [{u'}^{2}]$

Step 4.5

Combine the numerators over the common denominator.

${u'}^{2} (\frac{1}{2} {(2 u' - 9)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d v} [2 u' - 9]) + {(2 u' - 9)}^{\frac{1}{2}} \frac{d}{d v} [{u'}^{2}]$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 1$ by $2$ .

${u'}^{2} (\frac{1}{2} {(2 u' - 9)}^{\frac{1 - 2}{2}} \frac{d}{d v} [2 u' - 9]) + {(2 u' - 9)}^{\frac{1}{2}} \frac{d}{d v} [{u'}^{2}]$

Step 4.6.2

Subtract $2$ from $1$ .

${u'}^{2} (\frac{1}{2} {(2 u' - 9)}^{\frac{- 1}{2}} \frac{d}{d v} [2 u' - 9]) + {(2 u' - 9)}^{\frac{1}{2}} \frac{d}{d v} [{u'}^{2}]$

Step 4.7

Combine fractions.

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Step 4.7.1

Move the negative in front of the fraction.

${u'}^{2} (\frac{1}{2} {(2 u' - 9)}^{- \frac{1}{2}} \frac{d}{d v} [2 u' - 9]) + {(2 u' - 9)}^{\frac{1}{2}} \frac{d}{d v} [{u'}^{2}]$

Step 4.7.2

Combine $\frac{1}{2}$ and ${(2 u' - 9)}^{- \frac{1}{2}}$ .

${u'}^{2} (\frac{{(2 u' - 9)}^{- \frac{1}{2}}}{2} \frac{d}{d v} [2 u' - 9]) + {(2 u' - 9)}^{\frac{1}{2}} \frac{d}{d v} [{u'}^{2}]$

Step 4.7.3

Move ${(2 u' - 9)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${u'}^{2} (\frac{1}{2 {(2 u' - 9)}^{\frac{1}{2}}} \frac{d}{d v} [2 u' - 9]) + {(2 u' - 9)}^{\frac{1}{2}} \frac{d}{d v} [{u'}^{2}]$

Step 4.7.4

Combine $\frac{1}{2 {(2 u' - 9)}^{\frac{1}{2}}}$ and ${u'}^{2}$ .

$\frac{{u'}^{2}}{2 {(2 u' - 9)}^{\frac{1}{2}}} \frac{d}{d v} [2 u' - 9] + {(2 u' - 9)}^{\frac{1}{2}} \frac{d}{d v} [{u'}^{2}]$

Step 4.8

By the Sum Rule, the derivative of $2 u' - 9$ with respect to $u'$ is $\frac{d}{d v} [2 u'] + \frac{d}{d v} [- 9]$ .

$\frac{{u'}^{2}}{2 {(2 u' - 9)}^{\frac{1}{2}}} (\frac{d}{d v} [2 u'] + \frac{d}{d v} [- 9]) + {(2 u' - 9)}^{\frac{1}{2}} \frac{d}{d v} [{u'}^{2}]$

Step 4.9

Since $2$ is constant with respect to $u'$ , the derivative of $2 u'$ with respect to $u'$ is $2 \frac{d}{d v} [u']$ .

$\frac{{u'}^{2}}{2 {(2 u' - 9)}^{\frac{1}{2}}} (2 \frac{d}{d v} [u'] + \frac{d}{d v} [- 9]) + {(2 u' - 9)}^{\frac{1}{2}} \frac{d}{d v} [{u'}^{2}]$

Step 4.10

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

$\frac{{u'}^{2}}{2 {(2 u' - 9)}^{\frac{1}{2}}} (2 \cdot 1 + \frac{d}{d v} [- 9]) + {(2 u' - 9)}^{\frac{1}{2}} \frac{d}{d v} [{u'}^{2}]$

Step 4.11

Multiply $2$ by $1$ .

$\frac{{u'}^{2}}{2 {(2 u' - 9)}^{\frac{1}{2}}} (2 + \frac{d}{d v} [- 9]) + {(2 u' - 9)}^{\frac{1}{2}} \frac{d}{d v} [{u'}^{2}]$

Step 4.12

Since $- 9$ is constant with respect to $u'$ , the derivative of $- 9$ with respect to $u'$ is $0$ .

$\frac{{u'}^{2}}{2 {(2 u' - 9)}^{\frac{1}{2}}} (2 + 0) + {(2 u' - 9)}^{\frac{1}{2}} \frac{d}{d v} [{u'}^{2}]$

Step 4.13

Simplify terms.

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Step 4.13.1

Add $2$ and $0$ .

$\frac{{u'}^{2}}{2 {(2 u' - 9)}^{\frac{1}{2}}} \cdot 2 + {(2 u' - 9)}^{\frac{1}{2}} \frac{d}{d v} [{u'}^{2}]$

Step 4.13.2

Combine $\frac{{u'}^{2}}{2 {(2 u' - 9)}^{\frac{1}{2}}}$ and $2$ .

$\frac{{u'}^{2} \cdot 2}{2 {(2 u' - 9)}^{\frac{1}{2}}} + {(2 u' - 9)}^{\frac{1}{2}} \frac{d}{d v} [{u'}^{2}]$

Step 4.13.3

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

$\frac{2 \cdot {u'}^{2}}{2 {(2 u' - 9)}^{\frac{1}{2}}} + {(2 u' - 9)}^{\frac{1}{2}} \frac{d}{d v} [{u'}^{2}]$

Step 4.13.4

Cancel the common factor.

$\frac{2 {u'}^{2}}{2 {(2 u' - 9)}^{\frac{1}{2}}} + {(2 u' - 9)}^{\frac{1}{2}} \frac{d}{d v} [{u'}^{2}]$

Step 4.13.5

Rewrite the expression.

$\frac{{u'}^{2}}{{(2 u' - 9)}^{\frac{1}{2}}} + {(2 u' - 9)}^{\frac{1}{2}} \frac{d}{d v} [{u'}^{2}]$

Step 4.14

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

$\frac{{u'}^{2}}{{(2 u' - 9)}^{\frac{1}{2}}} + {(2 u' - 9)}^{\frac{1}{2}} (2 u')$

Step 4.15

Move $2$ to the left of ${(2 u' - 9)}^{\frac{1}{2}}$ .

$\frac{{u'}^{2}}{{(2 u' - 9)}^{\frac{1}{2}}} + 2 \cdot {(2 u' - 9)}^{\frac{1}{2}} u'$

Step 4.16

Combine $\frac{{u'}^{2}}{{(2 u' - 9)}^{\frac{1}{2}}}$ and $2 {(2 u' - 9)}^{\frac{1}{2}} u'$ using a common denominator.

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Step 4.16.1

Move ${(2 u' - 9)}^{\frac{1}{2}}$ .

$\frac{{u'}^{2}}{{(2 u' - 9)}^{\frac{1}{2}}} + 2 u' {(2 u' - 9)}^{\frac{1}{2}}$

Step 4.16.2

To write $2 u' {(2 u' - 9)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(2 u' - 9)}^{\frac{1}{2}}}{{(2 u' - 9)}^{\frac{1}{2}}}$ .

$\frac{{u'}^{2}}{{(2 u' - 9)}^{\frac{1}{2}}} + \frac{2 u' {(2 u' - 9)}^{\frac{1}{2}} {(2 u' - 9)}^{\frac{1}{2}}}{{(2 u' - 9)}^{\frac{1}{2}}}$

Step 4.16.3

Combine the numerators over the common denominator.

$\frac{{u'}^{2} + 2 u' {(2 u' - 9)}^{\frac{1}{2}} {(2 u' - 9)}^{\frac{1}{2}}}{{(2 u' - 9)}^{\frac{1}{2}}}$

Step 4.17

Multiply ${(2 u' - 9)}^{\frac{1}{2}}$ by ${(2 u' - 9)}^{\frac{1}{2}}$ by adding the exponents.

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Step 4.17.1

Move ${(2 u' - 9)}^{\frac{1}{2}}$ .

$\frac{{u'}^{2} + 2 u' ({(2 u' - 9)}^{\frac{1}{2}} {(2 u' - 9)}^{\frac{1}{2}})}{{(2 u' - 9)}^{\frac{1}{2}}}$

Step 4.17.2

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

$\frac{{u'}^{2} + 2 u' {(2 u' - 9)}^{\frac{1}{2} + \frac{1}{2}}}{{(2 u' - 9)}^{\frac{1}{2}}}$

Step 4.17.3

Combine the numerators over the common denominator.

$\frac{{u'}^{2} + 2 u' {(2 u' - 9)}^{\frac{1 + 1}{2}}}{{(2 u' - 9)}^{\frac{1}{2}}}$

Step 4.17.4

Add $1$ and $1$ .

$\frac{{u'}^{2} + 2 u' {(2 u' - 9)}^{\frac{2}{2}}}{{(2 u' - 9)}^{\frac{1}{2}}}$

Step 4.17.5

Divide $2$ by $2$ .

$\frac{{u'}^{2} + 2 u' {(2 u' - 9)}^{1}}{{(2 u' - 9)}^{\frac{1}{2}}}$

Step 4.18

Simplify $2 u' {(2 u' - 9)}^{1}$ .

$\frac{{u'}^{2} + 2 u' (2 u' - 9)}{{(2 u' - 9)}^{\frac{1}{2}}}$

Step 4.19

Simplify.

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Step 4.19.1

Apply the distributive property.

$\frac{{u'}^{2} + 2 u' (2 u') + 2 u' \cdot - 9}{{(2 u' - 9)}^{\frac{1}{2}}}$

Step 4.19.2

Simplify the numerator.

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Step 4.19.2.1

Simplify each term.

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Step 4.19.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{{u'}^{2} + 2 \cdot 2 u' u' + 2 u' \cdot - 9}{{(2 u' - 9)}^{\frac{1}{2}}}$

Step 4.19.2.1.2

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

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Step 4.19.2.1.2.1

Move $u'$ .

$\frac{{u'}^{2} + 2 \cdot 2 (u' u') + 2 u' \cdot - 9}{{(2 u' - 9)}^{\frac{1}{2}}}$

Step 4.19.2.1.2.2

Multiply $u'$ by $u'$ .

$\frac{{u'}^{2} + 2 \cdot 2 {u'}^{2} + 2 u' \cdot - 9}{{(2 u' - 9)}^{\frac{1}{2}}}$

Step 4.19.2.1.3

Multiply $2$ by $2$ .

$\frac{{u'}^{2} + 4 {u'}^{2} + 2 u' \cdot - 9}{{(2 u' - 9)}^{\frac{1}{2}}}$

Step 4.19.2.1.4

Multiply $- 9$ by $2$ .

$\frac{{u'}^{2} + 4 {u'}^{2} - 18 u'}{{(2 u' - 9)}^{\frac{1}{2}}}$

Step 4.19.2.2

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

$\frac{5 {u'}^{2} - 18 u'}{{(2 u' - 9)}^{\frac{1}{2}}}$

Step 4.19.3

Factor $u'$ out of $5 {u'}^{2} - 18 u'$ .

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Step 4.19.3.1

Factor $u'$ out of $5 {u'}^{2}$ .

$\frac{u' (5 u') - 18 u'}{{(2 u' - 9)}^{\frac{1}{2}}}$

Step 4.19.3.2

Factor $u'$ out of $- 18 u'$ .

$\frac{u' (5 u') + u' \cdot - 18}{{(2 u' - 9)}^{\frac{1}{2}}}$

Step 4.19.3.3

Factor $u'$ out of $u' (5 u') + u' \cdot - 18$ .

$\frac{u' (5 u' - 18)}{{(2 u' - 9)}^{\frac{1}{2}}}$

Step 5

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

$u' = \frac{u' (5 u' - 18)}{{(2 u' - 9)}^{\frac{1}{2}}}$

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

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