You are given 3 equations up front. If you take the first 2 equations and set them equal to each other, this accomplishes two things: first, doing this eliminates the variable y, and second, if the graphs of the first 2 equations intersect, at the point of intersection the y-value of one must equal the y-value of the other.
The third equation is easier to solve, because the variable y has been eliminated, leaving only x as variable.
Please take a look at the remainder of this question and see what you yourself are able to do. Share your work. Then, if you'd message me, I'd be glad to take another look and give you suggestions.
The pythagorean theorem is A squared plus B squared equals C squared. and it is used to find a missing length in a right triangle:)
hope this helps!
Answer:
![\displaystyle \frac{dy}{dx} \bigg| \limit_{(1, 4)} = 2](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bdy%7D%7Bdx%7D%20%5Cbigg%7C%20%5Climit_%7B%281%2C%204%29%7D%20%3D%202)
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>
- Coordinates (x, y)
- 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)
- 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)
<u>Calculus</u>
Derivatives
Derivative Notation
Derivative of a constant is 0
Implicit Differentiation
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Step-by-step explanation:
<u>Step 1: Define</u>
![\displaystyle \sqrt{x} - \sqrt{y} = -1](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Csqrt%7Bx%7D%20-%20%5Csqrt%7By%7D%20%3D%20-1)
Point (1, 4)
<u>Step 2: Differentiate</u>
- [Function] Rewrite [Exponential Rule - Root Rewrite]:
![\displaystyle x^{\frac{1}{2}} - y^{\frac{1}{2}} = -1](https://tex.z-dn.net/?f=%5Cdisplaystyle%20x%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%20-%20y%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%20%3D%20-1)
- [Implicit Differentiation] 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-%20%5Cfrac%7B1%7D%7B2%7Dy%5E%7B%5Cfrac%7B1%7D%7B2%7D%20-%201%7D%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%200)
- [Implicit Differentiation] Simplify Exponents:
![\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-%20%5Cfrac%7B1%7D%7B2%7Dy%5E%7B%5Cfrac%7B-1%7D%7B2%7D%7D%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%200)
- [Implicit Differentiation] 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-%20%5Cfrac%7B1%7D%7B2y%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%7D%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%200)
- [Implicit Differentiation] Isolate <em>y</em> terms:
![\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)
- [Implicit Differentiation] Isolate
: ![\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)
- [Implicit Differentiation] 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>
- Substitute in point [Derivative]:
![\displaystyle \frac{dy}{dx} = \frac{(4)^{\frac{1}{2}}}{(1)^{\frac{1}{2}}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%20%5Cfrac%7B%284%29%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%7D%7B%281%29%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%7D)
- Exponents:
![\displaystyle \frac{dy}{dx} = \frac{2}{1}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%20%5Cfrac%7B2%7D%7B1%7D)
- Division:
![\displaystyle \frac{dy}{dx} = 2](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%202)
Topic: AP Calculus AB/BC (Calculus I/II)
Unit: Implicit Differentiation
Book: College Calculus 10e
Answer:
Step-by-step explanation:
57/100 + 3/10
they need common denominators so a common denominator is 100
so 10x10=100 and 3x10= 30 so we now got
57/100 + 30/100 which is 87/100