Step-by-step explanation:
what was love What was this like? and who do you love
Answer:
we need a picture, or the rectangles
Answer:
![y''(-1) =8](https://tex.z-dn.net/?f=y%27%27%28-1%29%20%3D8)
General Formulas and Concepts:
<u>Pre-Algebra</u>
Order of Operations: BPEMDAS
- Brackets
- Parenthesis
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
Equality Properties
<u>Algebra I</u>
<u>Calculus</u>
Implicit Differentiation
The derivative of a constant is equal to 0
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Product Rule: ![\frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)](https://tex.z-dn.net/?f=%5Cfrac%7Bd%7D%7Bdx%7D%20%5Bf%28x%29g%28x%29%5D%3Df%27%28x%29g%28x%29%20%2B%20g%27%28x%29f%28x%29)
Chain Rule: ![\frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)](https://tex.z-dn.net/?f=%5Cfrac%7Bd%7D%7Bdx%7D%5Bf%28g%28x%29%29%5D%20%3Df%27%28g%28x%29%29%20%5Ccdot%20g%27%28x%29)
Quotient Rule: ![\frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}](https://tex.z-dn.net/?f=%5Cfrac%7Bd%7D%7Bdx%7D%20%5B%5Cfrac%7Bf%28x%29%7D%7Bg%28x%29%7D%20%5D%3D%5Cfrac%7Bg%28x%29f%27%28x%29-g%27%28x%29f%28x%29%7D%7Bg%5E2%28x%29%7D)
Step-by-step explanation:
<u>Step 1: Define</u>
-xy - 2y = -4
Rate of change of the tangent line at point (-1, 4)
<u>Step 2: Differentiate Pt. 1</u>
<em>Find 1st Derivative</em>
- Implicit Differentiation [Product Rule/Basic Power Rule]:
![-y - xy' - 2y' = 0](https://tex.z-dn.net/?f=-y%20-%20xy%27%20-%202y%27%20%3D%200)
- [Algebra] Isolate <em>y'</em> terms:
![-xy' - 2y' = y](https://tex.z-dn.net/?f=-xy%27%20-%202y%27%20%3D%20y)
- [Algebra] Factor <em>y'</em>:
![y'(-x - 2) = y](https://tex.z-dn.net/?f=y%27%28-x%20-%202%29%20%3D%20y)
- [Algebra] Isolate <em>y'</em>:
![y' = \frac{y}{-x-2}](https://tex.z-dn.net/?f=y%27%20%3D%20%5Cfrac%7By%7D%7B-x-2%7D)
- [Algebra] Rewrite:
![y' = \frac{-y}{x+2}](https://tex.z-dn.net/?f=y%27%20%3D%20%5Cfrac%7B-y%7D%7Bx%2B2%7D)
<u>Step 3: Find </u><em><u>y</u></em>
- Define equation:
![-xy - 2y = -4](https://tex.z-dn.net/?f=-xy%20-%202y%20%3D%20-4)
- Factor <em>y</em>:
![y(-x - 2) = -4](https://tex.z-dn.net/?f=y%28-x%20-%202%29%20%3D%20-4)
- Isolate <em>y</em>:
![y = \frac{-4}{-x-2}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B-4%7D%7B-x-2%7D)
- Simplify:
![y = \frac{4}{x+2}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B4%7D%7Bx%2B2%7D)
<u>Step 4: Rewrite 1st Derivative</u>
- [Algebra] Substitute in <em>y</em>:
![y' = \frac{-\frac{4}{x+2} }{x+2}](https://tex.z-dn.net/?f=y%27%20%3D%20%5Cfrac%7B-%5Cfrac%7B4%7D%7Bx%2B2%7D%20%7D%7Bx%2B2%7D)
- [Algebra] Simplify:
![y' = \frac{-4}{(x+2)^2}](https://tex.z-dn.net/?f=y%27%20%3D%20%5Cfrac%7B-4%7D%7B%28x%2B2%29%5E2%7D)
<u>Step 5: Differentiate Pt. 2</u>
<em>Find 2nd Derivative</em>
- Differentiate [Quotient Rule/Basic Power Rule]:
![y'' = \frac{0(x+2)^2 - 8 \cdot 2(x + 2) \cdot 1}{[(x + 2)^2]^2}](https://tex.z-dn.net/?f=y%27%27%20%3D%20%5Cfrac%7B0%28x%2B2%29%5E2%20-%208%20%5Ccdot%202%28x%20%2B%202%29%20%5Ccdot%201%7D%7B%5B%28x%20%2B%202%29%5E2%5D%5E2%7D)
- [Derivative] Simplify:
![y'' = \frac{8}{(x+2)^3}](https://tex.z-dn.net/?f=y%27%27%20%3D%20%5Cfrac%7B8%7D%7B%28x%2B2%29%5E3%7D)
<u>Step 6: Find Slope at Given Point</u>
- [Algebra] Substitute in <em>x</em>:
![y''(-1) = \frac{8}{(-1+2)^3}](https://tex.z-dn.net/?f=y%27%27%28-1%29%20%3D%20%5Cfrac%7B8%7D%7B%28-1%2B2%29%5E3%7D)
- [Algebra] Evaluate:
![y''(-1) =8](https://tex.z-dn.net/?f=y%27%27%28-1%29%20%3D8)
Answer:
1
Step-by-step explanation:
Anything to the power of 0 equals 1
Answer:
No solution is possible from the provided information
Step-by-step explanation: