6x to the power of 2
6x^2
Answer: 36x
Answer:
<h3>11 teams having equal 9 members.</h3><h3>The number of members in a team is 9</h3>
Step-by-step explanation:
Given a club has 54 male members and 45 female members. Also given that the club is having a contest, and there must be are all male and all female in teams competing the contest.
<h3>To find the most members a team ca have :</h3>
Male =54 members and Female =45 members
Total number of members = male members + female members
=54+45
=99
∴ Total number of members=99
∴ The number of members in a team is 9.
From the given problem the teams must all be equal, and each and every member should be on a team.
<h3>Hence 11 teams equals 9 members</h3>
Answer:
y = 4
Step-by-step explanation:
4*y = 4y
4*4 = 16
4y+16=32
to get 4y alone, we subtract 16 from 32.
32-16 = 16
4y = 16
to get y alone, we divide 4 by 16
16/ 4 = 4
so, y = 4
Answer:
B. ![\displaystyle -\frac{1}{3}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20-%5Cfrac%7B1%7D%7B3%7D)
General Formulas and Concepts:
<u>Pre-Algebra</u>
Order of Operations: BPEMDAS
- Brackets
- Parenthesis
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
Equality Properties
- Multiplication Property of Equality
- Division Property of Equality
- Addition Property of Equality
- Subtraction Property of Equality
<u>Algebra I</u>
- Terms/Coefficients
- Exponential Rule [Rewrite]:
![\displaystyle b^{-m} = \frac{1}{b^m}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20b%5E%7B-m%7D%20%3D%20%5Cfrac%7B1%7D%7Bb%5Em%7D)
- Exponential Rule [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>
Derivatives
Derivative Notation
Derivative of a constant is 0
The definition of a derivative is the slope of the tangent line
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Implicit Differentiation
Step-by-step explanation:
<u>Step 1: Define</u>
![\displaystyle \sqrt{x} + \sqrt{y} = 2](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Csqrt%7Bx%7D%20%2B%20%5Csqrt%7By%7D%20%3D%202)
![\displaystyle (\frac{9}{4}, \frac{1}{4})](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%28%5Cfrac%7B9%7D%7B4%7D%2C%20%5Cfrac%7B1%7D%7B4%7D%29)
<u>Step 2: Differentiate</u>
<em>Implicit Differentiation</em>
- [Function] Rewrite [Exponential Rule - Root Rewrite]:
![\displaystyle x^{\frac{1}{2}} + y^{\frac{1}{2}} = 2](https://tex.z-dn.net/?f=%5Cdisplaystyle%20x%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%20%2B%20y%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%20%3D%202)
- [Function] Basic Power Rule:
![\displaystyle \frac{1}{2}x^{\frac{1}{2} - 1} + \frac{1}{2}y^{\frac{1}{2} - 1}\frac{dy}{dx} = 0](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B1%7D%7B2%7Dx%5E%7B%5Cfrac%7B1%7D%7B2%7D%20-%201%7D%20%2B%20%5Cfrac%7B1%7D%7B2%7Dy%5E%7B%5Cfrac%7B1%7D%7B2%7D%20-%201%7D%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%200)
- [Derivative] Simplify:
![\displaystyle \frac{1}{2}x^{\frac{-1}{2}} + \frac{1}{2}y^{\frac{-1}{2}}\frac{dy}{dx} = 0](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B1%7D%7B2%7Dx%5E%7B%5Cfrac%7B-1%7D%7B2%7D%7D%20%2B%20%5Cfrac%7B1%7D%7B2%7Dy%5E%7B%5Cfrac%7B-1%7D%7B2%7D%7D%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%200)
- [Derivative] Rewrite [Exponential Rule - Rewrite]:
![\displaystyle \frac{1}{2x^{\frac{1}{2}}} + \frac{1}{2y^{\frac{1}{2}}}\frac{dy}{dx} = 0](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B1%7D%7B2x%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%7D%20%2B%20%5Cfrac%7B1%7D%7B2y%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%7D%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%200)
- [Derivative] Isolate
term [Subtraction Property of Equality]: ![\displaystyle \frac{1}{2y^{\frac{1}{2}}}\frac{dy}{dx} = -\frac{1}{2x^{\frac{1}{2}}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B1%7D%7B2y%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%7D%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%20-%5Cfrac%7B1%7D%7B2x%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%7D)
- [Derivative] Isolate
[Multiplication Property of Equality]: ![\displaystyle \frac{dy}{dx} = -\frac{2y^{\frac{1}{2}}}{2x^{\frac{1}{2}}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%20-%5Cfrac%7B2y%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%7D%7B2x%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%7D)
- [Derivative] Simplify:
![\displaystyle \frac{dy}{dx} = -\frac{y^{\frac{1}{2}}}{x^{\frac{1}{2}}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%20-%5Cfrac%7By%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%7D%7Bx%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%7D)
<u>Step 3: Evaluate</u>
<em>Find slope of tangent line</em>
- Substitute in point [Derivative]:
![\displaystyle \frac{dy}{dx} \bigg| \limit_{(\frac{9}{4}, \frac{1}{4})} = -\frac{(\frac{1}{4})^{\frac{1}{2}}}{(\frac{9}{4})^{\frac{1}{2}}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bdy%7D%7Bdx%7D%20%5Cbigg%7C%20%5Climit_%7B%28%5Cfrac%7B9%7D%7B4%7D%2C%20%5Cfrac%7B1%7D%7B4%7D%29%7D%20%3D%20-%5Cfrac%7B%28%5Cfrac%7B1%7D%7B4%7D%29%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%7D%7B%28%5Cfrac%7B9%7D%7B4%7D%29%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%7D)
- [Slope] Exponents:
![\displaystyle \frac{dy}{dx} \bigg| \limit_{(\frac{9}{4}, \frac{1}{4})} = -\frac{\frac{1}{2}}{\frac{3}{2}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bdy%7D%7Bdx%7D%20%5Cbigg%7C%20%5Climit_%7B%28%5Cfrac%7B9%7D%7B4%7D%2C%20%5Cfrac%7B1%7D%7B4%7D%29%7D%20%3D%20-%5Cfrac%7B%5Cfrac%7B1%7D%7B2%7D%7D%7B%5Cfrac%7B3%7D%7B2%7D%7D)
- [Slope] Simplify:
![\displaystyle \frac{dy}{dx} \bigg| \limit_{(\frac{9}{4}, \frac{1}{4})} = -\frac{1}{3}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bdy%7D%7Bdx%7D%20%5Cbigg%7C%20%5Climit_%7B%28%5Cfrac%7B9%7D%7B4%7D%2C%20%5Cfrac%7B1%7D%7B4%7D%29%7D%20%3D%20-%5Cfrac%7B1%7D%7B3%7D)
Topic: AP Calculus AB/BC (Calculus I/II)
Unit: Differentiation - Implicit Differentiation
Book: College Calculus 10e