Calculus Examples

$g (u) = \sqrt{2} u + \sqrt{3 u}$

Step 1

By the Sum Rule, the derivative of $\sqrt{2} u + \sqrt{3 u}$ with respect to $u$ is $\frac{d}{d u} [\sqrt{2} u] + \frac{d}{d u} [\sqrt{3 u}]$ .

$\frac{d}{d u} [\sqrt{2} u] + \frac{d}{d u} [\sqrt{3 u}]$

Step 2

Evaluate $\frac{d}{d u} [\sqrt{2} u]$ .

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Step 2.1

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

$\sqrt{2} \frac{d}{d u} [u] + \frac{d}{d u} [\sqrt{3 u}]$

Step 2.2

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

$\sqrt{2} \cdot 1 + \frac{d}{d u} [\sqrt{3 u}]$

Step 2.3

Multiply $\sqrt{2}$ by $1$ .

$\sqrt{2} + \frac{d}{d u} [\sqrt{3 u}]$

Step 3

Evaluate $\frac{d}{d u} [\sqrt{3 u}]$ .

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Step 3.1

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

$\sqrt{2} + \frac{d}{d u} [{(3 u)}^{\frac{1}{2}}]$

Step 3.2

Factor $3$ out of $3 u$ .

$\sqrt{2} + \frac{d}{d u} [{(3 (u))}^{\frac{1}{2}}]$

Step 3.3

Apply the product rule to $3 (u)$ .

$\sqrt{2} + \frac{d}{d u} [3^{\frac{1}{2}} u^{\frac{1}{2}}]$

Step 3.4

Since $3^{\frac{1}{2}}$ is constant with respect to $u$ , the derivative of $3^{\frac{1}{2}} u^{\frac{1}{2}}$ with respect to $u$ is $3^{\frac{1}{2}} \frac{d}{d u} [u^{\frac{1}{2}}]$ .

$\sqrt{2} + 3^{\frac{1}{2}} \frac{d}{d u} [u^{\frac{1}{2}}]$

Step 3.5

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

$\sqrt{2} + 3^{\frac{1}{2}} (\frac{1}{2} u^{\frac{1}{2} - 1})$

Step 3.6

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

$\sqrt{2} + 3^{\frac{1}{2}} (\frac{1}{2} u^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 3.7

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

$\sqrt{2} + 3^{\frac{1}{2}} (\frac{1}{2} u^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 3.8

Combine the numerators over the common denominator.

$\sqrt{2} + 3^{\frac{1}{2}} (\frac{1}{2} u^{\frac{1 - 1 \cdot 2}{2}})$

Step 3.9

Simplify the numerator.

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Step 3.9.1

Multiply $- 1$ by $2$ .

$\sqrt{2} + 3^{\frac{1}{2}} (\frac{1}{2} u^{\frac{1 - 2}{2}})$

Step 3.9.2

Subtract $2$ from $1$ .

$\sqrt{2} + 3^{\frac{1}{2}} (\frac{1}{2} u^{\frac{- 1}{2}})$

Step 3.10

Move the negative in front of the fraction.

$\sqrt{2} + 3^{\frac{1}{2}} (\frac{1}{2} u^{- \frac{1}{2}})$

Step 3.11

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

$\sqrt{2} + 3^{\frac{1}{2}} \frac{u^{- \frac{1}{2}}}{2}$

Step 3.12

Combine $3^{\frac{1}{2}}$ and $\frac{u^{- \frac{1}{2}}}{2}$ .

$\sqrt{2} + \frac{3^{\frac{1}{2}} u^{- \frac{1}{2}}}{2}$

Step 3.13

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

$\sqrt{2} + \frac{3^{\frac{1}{2}}}{2 u^{\frac{1}{2}}}$

Step 4

Reorder terms.

$\frac{3^{\frac{1}{2}}}{2 u^{\frac{1}{2}}} + \sqrt{2}$