Answer:
![\displaystyle \frac{d}{dx} \bigg[ \frac{1}{\sqrt{4x}} \bigg] = \frac{-1}{4x^\bigg{\frac{3}{2}}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5Cbigg%5B%20%5Cfrac%7B1%7D%7B%5Csqrt%7B4x%7D%7D%20%5Cbigg%5D%20%3D%20%5Cfrac%7B-1%7D%7B4x%5E%5Cbigg%7B%5Cfrac%7B3%7D%7B2%7D%7D%7D)
General Formulas and Concepts:
<u>Pre-Algebra</u>
Order of Operations: BPEMDAS
- Brackets
- Parenthesis
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
<u>Algebra I</u>
Exponential Properties
- Exponential Property [Rewrite]:

- Exponential Property [Root Rewrite]:
![\displaystyle \sqrt[n]{x} = x^{\frac{1}{n}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Csqrt%5Bn%5D%7Bx%7D%20%3D%20x%5E%7B%5Cfrac%7B1%7D%7Bn%7D%7D)
<u>Calculus</u>
Differentiation
- Derivatives
- Derivative Notation
Derivative Property [Multiplied Constant]:
Derivative Rule [Basic Power Rule]:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Step-by-step explanation:
<u>Step 1: Define</u>
<em>Identify.</em>
![\displaystyle \frac{d}{dx} \bigg[ \frac{1}{\sqrt{4x}} \bigg]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5Cbigg%5B%20%5Cfrac%7B1%7D%7B%5Csqrt%7B4x%7D%7D%20%5Cbigg%5D)
<u>Step 2: Differentiate</u>
- Simplify:
![\displaystyle \frac{d}{dx} \bigg[ \frac{1}{\sqrt{4x}} \bigg] = \bigg( \frac{1}{2\sqrt{x}} \bigg)'](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5Cbigg%5B%20%5Cfrac%7B1%7D%7B%5Csqrt%7B4x%7D%7D%20%5Cbigg%5D%20%3D%20%5Cbigg%28%20%5Cfrac%7B1%7D%7B2%5Csqrt%7Bx%7D%7D%20%5Cbigg%29%27)
- Rewrite [Derivative Property - Multiplied Constant]:
![\displaystyle \frac{d}{dx} \bigg[ \frac{1}{\sqrt{4x}} \bigg] = \frac{1}{2} \bigg( \frac{1}{\sqrt{x}} \bigg)'](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5Cbigg%5B%20%5Cfrac%7B1%7D%7B%5Csqrt%7B4x%7D%7D%20%5Cbigg%5D%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20%5Cbigg%28%20%5Cfrac%7B1%7D%7B%5Csqrt%7Bx%7D%7D%20%5Cbigg%29%27)
- Rewrite [Exponential Rule - Root Rewrite]:
![\displaystyle \frac{d}{dx} \bigg[ \frac{1}{\sqrt{4x}} \bigg] = \frac{1}{2} \bigg( \frac{1}{x^\Big{\frac{1}{2}}} \bigg)'](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5Cbigg%5B%20%5Cfrac%7B1%7D%7B%5Csqrt%7B4x%7D%7D%20%5Cbigg%5D%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20%5Cbigg%28%20%5Cfrac%7B1%7D%7Bx%5E%5CBig%7B%5Cfrac%7B1%7D%7B2%7D%7D%7D%20%5Cbigg%29%27)
- Rewrite [Exponential Rule - Rewrite]:
![\displaystyle \frac{d}{dx} \bigg[ \frac{1}{\sqrt{4x}} \bigg] = \frac{1}{2} \bigg( x^\bigg{\frac{-1}{2}} \bigg)'](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5Cbigg%5B%20%5Cfrac%7B1%7D%7B%5Csqrt%7B4x%7D%7D%20%5Cbigg%5D%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20%5Cbigg%28%20x%5E%5Cbigg%7B%5Cfrac%7B-1%7D%7B2%7D%7D%20%5Cbigg%29%27)
- Derivative Rule [Basic Power Rule]:
![\displaystyle \frac{d}{dx} \bigg[ \frac{1}{\sqrt{4x}} \bigg] = \frac{1}{2} \bigg( \frac{-1}{2} x^\bigg{\frac{-3}{2}} \bigg)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5Cbigg%5B%20%5Cfrac%7B1%7D%7B%5Csqrt%7B4x%7D%7D%20%5Cbigg%5D%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20%5Cbigg%28%20%5Cfrac%7B-1%7D%7B2%7D%20x%5E%5Cbigg%7B%5Cfrac%7B-3%7D%7B2%7D%7D%20%5Cbigg%29)
- Simplify:
![\displaystyle \frac{d}{dx} \bigg[ \frac{1}{\sqrt{4x}} \bigg] = \frac{-1}{4} x^\bigg{\frac{-3}{2}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5Cbigg%5B%20%5Cfrac%7B1%7D%7B%5Csqrt%7B4x%7D%7D%20%5Cbigg%5D%20%3D%20%5Cfrac%7B-1%7D%7B4%7D%20x%5E%5Cbigg%7B%5Cfrac%7B-3%7D%7B2%7D%7D)
- Rewrite [Exponential Rule - Rewrite]:
![\displaystyle \frac{d}{dx} \bigg[ \frac{1}{\sqrt{4x}} \bigg] = \frac{-1}{4x^\bigg{\frac{3}{2}}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5Cbigg%5B%20%5Cfrac%7B1%7D%7B%5Csqrt%7B4x%7D%7D%20%5Cbigg%5D%20%3D%20%5Cfrac%7B-1%7D%7B4x%5E%5Cbigg%7B%5Cfrac%7B3%7D%7B2%7D%7D%7D)
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
1) 2:5 ratio
This is a fairly simple process, you count how many of each and then put them in order the way the question asks.
So there are 5 cans and 2 bottles. The question asks "bottles to cans" so the correct answer HAS to be 2:5, NOT 5:2.
2) 3:1
There is 1 spoon and 3 forks. The question asks "forks to spoons" so the correct answer HAS to be 3:1, NOT 1:3
I am taking this quiz but I'm pretty sure its....c
Answer:
It will take her 11.4 days, or 12 whole days to jog 57 miles
Step-by-step explanation:
1 day = 5 miles
? days = 57 miles
57/5 = 11.4 days
We need to solve the net volume of the given problem and we have 6 total rectangles with areas. Solving for the volume we have:
Length = 6 yard
width = 4 yard
height = 2 yard
Solving for the volume, we have:
Volume = 6 x 4 x 2
Volume = 48 yard³
The answer is 48 yard³.