Calculus Examples

$- \sqrt{5 h}$

Step 1

Simplify with factoring out.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 h}$ as ${(5 h)}^{\frac{1}{2}}$ .

$\frac{d}{d h} [- {(5 h)}^{\frac{1}{2}}]$

Step 1.2

Factor $5$ out of $5 h$ .

$\frac{d}{d h} [- {(5 (h))}^{\frac{1}{2}}]$

Step 1.3

Apply the product rule to $5 (h)$ .

$\frac{d}{d h} [- (5^{\frac{1}{2}} h^{\frac{1}{2}})]$

Step 2

Since $- 5^{\frac{1}{2}}$ is constant with respect to $h$ , the derivative of $- 5^{\frac{1}{2}} h^{\frac{1}{2}}$ with respect to $h$ is $- 5^{\frac{1}{2}} \frac{d}{d h} [h^{\frac{1}{2}}]$ .

$- 5^{\frac{1}{2}} \frac{d}{d h} [h^{\frac{1}{2}}]$

Step 3

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

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

Step 4

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

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

Step 5

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

$- 5^{\frac{1}{2}} (\frac{1}{2} h^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Move the negative in front of the fraction.

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

Step 9

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

$- 5^{\frac{1}{2}} \frac{h^{- \frac{1}{2}}}{2}$

Step 10

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

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

Step 11

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

$- \frac{5^{\frac{1}{2}}}{2 h^{\frac{1}{2}}}$