Algebra Examples

$h = 33 (10 \sqrt{v} - v + 10.45)$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{h'}$ as ${h'}^{\frac{1}{2}}$ .

$h = 33 (10 {h'}^{\frac{1}{2}} - h' + 10.45)$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d h'} h = \frac{d}{d h'} (33 (10 {h'}^{\frac{1}{2}} - h' + 10.45))$

Step 3

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

$h'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Since $33$ is constant with respect to $h'$ , the derivative of $33 (10 {h'}^{\frac{1}{2}} - h' + 10.45)$ with respect to $h'$ is $33 \frac{d}{d v} [10 {h'}^{\frac{1}{2}} - h' + 10.45]$ .

$33 \frac{d}{d v} [10 {h'}^{\frac{1}{2}} - h' + 10.45]$

Step 4.2

By the Sum Rule, the derivative of $10 {h'}^{\frac{1}{2}} - h' + 10.45$ with respect to $h'$ is $\frac{d}{d v} [10 {h'}^{\frac{1}{2}}] + \frac{d}{d v} [- h'] + \frac{d}{d v} [10.45]$ .

$33 (\frac{d}{d v} [10 {h'}^{\frac{1}{2}}] + \frac{d}{d v} [- h'] + \frac{d}{d v} [10.45])$

Step 4.3

Since $10$ is constant with respect to $h'$ , the derivative of $10 {h'}^{\frac{1}{2}}$ with respect to $h'$ is $10 \frac{d}{d v} [{h'}^{\frac{1}{2}}]$ .

$33 (10 \frac{d}{d v} [{h'}^{\frac{1}{2}}] + \frac{d}{d v} [- h'] + \frac{d}{d v} [10.45])$

Step 4.4

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

$33 (10 (\frac{1}{2} {h'}^{\frac{1}{2} - 1}) + \frac{d}{d v} [- h'] + \frac{d}{d v} [10.45])$

Step 4.5

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

$33 (10 (\frac{1}{2} {h'}^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d v} [- h'] + \frac{d}{d v} [10.45])$

Step 4.6

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

$33 (10 (\frac{1}{2} {h'}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d v} [- h'] + \frac{d}{d v} [10.45])$

Step 4.7

Combine the numerators over the common denominator.

$33 (10 (\frac{1}{2} {h'}^{\frac{1 - 1 \cdot 2}{2}}) + \frac{d}{d v} [- h'] + \frac{d}{d v} [10.45])$

Step 4.8

Simplify the numerator.

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Step 4.8.1

Multiply $- 1$ by $2$ .

$33 (10 (\frac{1}{2} {h'}^{\frac{1 - 2}{2}}) + \frac{d}{d v} [- h'] + \frac{d}{d v} [10.45])$

Step 4.8.2

Subtract $2$ from $1$ .

$33 (10 (\frac{1}{2} {h'}^{\frac{- 1}{2}}) + \frac{d}{d v} [- h'] + \frac{d}{d v} [10.45])$

Step 4.9

Move the negative in front of the fraction.

$33 (10 (\frac{1}{2} {h'}^{- \frac{1}{2}}) + \frac{d}{d v} [- h'] + \frac{d}{d v} [10.45])$

Step 4.10

Combine $\frac{1}{2}$ and ${h'}^{- \frac{1}{2}}$ .

$33 (10 \frac{{h'}^{- \frac{1}{2}}}{2} + \frac{d}{d v} [- h'] + \frac{d}{d v} [10.45])$

Step 4.11

Combine $10$ and $\frac{{h'}^{- \frac{1}{2}}}{2}$ .

$33 (\frac{10 {h'}^{- \frac{1}{2}}}{2} + \frac{d}{d v} [- h'] + \frac{d}{d v} [10.45])$

Step 4.12

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

$33 (\frac{10}{2 {h'}^{\frac{1}{2}}} + \frac{d}{d v} [- h'] + \frac{d}{d v} [10.45])$

Step 4.13

Factor $2$ out of $10$ .

$33 (\frac{2 \cdot 5}{2 {h'}^{\frac{1}{2}}} + \frac{d}{d v} [- h'] + \frac{d}{d v} [10.45])$

Step 4.14

Cancel the common factors.

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Step 4.14.1

Factor $2$ out of $2 {h'}^{\frac{1}{2}}$ .

$33 (\frac{2 \cdot 5}{2 {h'}^{\frac{1}{2}}} + \frac{d}{d v} [- h'] + \frac{d}{d v} [10.45])$

Step 4.14.2

Cancel the common factor.

$33 (\frac{2 \cdot 5}{2 {h'}^{\frac{1}{2}}} + \frac{d}{d v} [- h'] + \frac{d}{d v} [10.45])$

Step 4.14.3

Rewrite the expression.

$33 (\frac{5}{{h'}^{\frac{1}{2}}} + \frac{d}{d v} [- h'] + \frac{d}{d v} [10.45])$

Step 4.15

Since $- 1$ is constant with respect to $h'$ , the derivative of $- h'$ with respect to $h'$ is $- \frac{d}{d v} [h']$ .

$33 (\frac{5}{{h'}^{\frac{1}{2}}} - \frac{d}{d v} [h'] + \frac{d}{d v} [10.45])$

Step 4.16

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

$33 (\frac{5}{{h'}^{\frac{1}{2}}} - 1 \cdot 1 + \frac{d}{d v} [10.45])$

Step 4.17

Multiply $- 1$ by $1$ .

$33 (\frac{5}{{h'}^{\frac{1}{2}}} - 1 + \frac{d}{d v} [10.45])$

Step 4.18

Since $10.45$ is constant with respect to $h'$ , the derivative of $10.45$ with respect to $h'$ is $0$ .

$33 (\frac{5}{{h'}^{\frac{1}{2}}} - 1 + 0)$

Step 4.19

Add $\frac{5}{{h'}^{\frac{1}{2}}} - 1$ and $0$ .

$33 (\frac{5}{{h'}^{\frac{1}{2}}} - 1)$

Step 4.20

Simplify.

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Step 4.20.1

Apply the distributive property.

$33 \frac{5}{{h'}^{\frac{1}{2}}} + 33 \cdot - 1$

Step 4.20.2

Combine terms.

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Step 4.20.2.1

Combine $33$ and $\frac{5}{{h'}^{\frac{1}{2}}}$ .

$\frac{33 \cdot 5}{{h'}^{\frac{1}{2}}} + 33 \cdot - 1$

Step 4.20.2.2

Multiply $33$ by $5$ .

$\frac{165}{{h'}^{\frac{1}{2}}} + 33 \cdot - 1$

Step 4.20.2.3

Multiply $33$ by $- 1$ .

$\frac{165}{{h'}^{\frac{1}{2}}} - 33$

Step 5

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

$h' = \frac{165}{{h'}^{\frac{1}{2}}} - 33$

Step 6

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

$h' \approx 12.9143206$

Step 7

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

$h' = \frac{d h}{d v} \approx 12.9143206$