Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Combine the numerators over the common denominator.

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

Step 9

Simplify the numerator.

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Step 9.1

Multiply $- 1$ by $2$ .

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

Step 9.2

Subtract $2$ from $1$ .

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

Step 10

Move the negative in front of the fraction.

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

Step 11

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

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

Step 12

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

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

Step 13

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

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

Step 14

Factor $2$ out of $10$ .

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

Step 15

Cancel the common factors.

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Step 15.1

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

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

Step 15.2

Cancel the common factor.

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

Step 15.3

Rewrite the expression.

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

Step 16

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

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

Step 17

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

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

Step 18

Multiply $- 1$ by $1$ .

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

Step 19

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

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

Step 20

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

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

Step 21

Simplify.

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Step 21.1

Apply the distributive property.

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

Step 21.2

Combine terms.

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Step 21.2.1

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

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

Step 21.2.2

Multiply $33$ by $5$ .

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

Step 21.2.3

Multiply $33$ by $- 1$ .

$\frac{165}{v^{\frac{1}{2}}} - 33$