Answer:

General Formulas and Concepts:
<u>Algebra I</u>
- Terms/Coefficients
- Functions
- Function Notation
- Factoring
<u>Calculus</u>
Derivatives
Derivative Notation
Derivative Property [Addition/Subtraction]: ![\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bf%28x%29%20%2B%20g%28x%29%5D%20%3D%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bf%28x%29%5D%20%2B%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bg%28x%29%5D)
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Derivative Rule [Product Rule]: ![\displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%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)
Derivative Rule [Chain Rule]: ![\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bf%28g%28x%29%29%5D%20%3Df%27%28g%28x%29%29%20%5Ccdot%20g%27%28x%29)
Explanation:
<u>Step 1: Define</u>
<em>Identify</em>
y = x(1 + x)³
<u>Step 2: Differentiate</u>
- Product Rule [Derivative Rule - Chain Rule]:
![\displaystyle y' = \frac{d}{dx}[x] \cdot (1 + x)^3 + x \cdot \frac{d}{dx}[(1 + x)^3] \cdot \frac{d}{dx}[1 + x]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bx%5D%20%5Ccdot%20%281%20%2B%20x%29%5E3%20%2B%20x%20%5Ccdot%20%5Cfrac%7Bd%7D%7Bdx%7D%5B%281%20%2B%20x%29%5E3%5D%20%5Ccdot%20%5Cfrac%7Bd%7D%7Bdx%7D%5B1%20%2B%20x%5D)
- Derivative Property [Addition/Subtraction]:
![\displaystyle y' = \frac{d}{dx}[x] \cdot (1 + x)^3 + x \cdot \frac{d}{dx}[(1 + x)^3] \cdot (\frac{d}{dx}[1] + \frac{d}{dx}[x])](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bx%5D%20%5Ccdot%20%281%20%2B%20x%29%5E3%20%2B%20x%20%5Ccdot%20%5Cfrac%7Bd%7D%7Bdx%7D%5B%281%20%2B%20x%29%5E3%5D%20%5Ccdot%20%28%5Cfrac%7Bd%7D%7Bdx%7D%5B1%5D%20%2B%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bx%5D%29)
- Basic Power Rule:

- Simplify:

- Factor:
![\displaystyle y' = (1 + x)^2 \bigg[ (1 + x) + 3x \bigg]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%20%281%20%2B%20x%29%5E2%20%5Cbigg%5B%20%281%20%2B%20x%29%20%2B%203x%20%5Cbigg%5D)
- Combine like terms:

Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Derivatives
Book: College Calculus 10e
Answer:
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As the information given in the question, the university must ask the researcher about the collaboration of the research at the research site in the abroad. The survey must include the individual from the different states and countries. All these surveys must be complied at the different research site present in abroad.
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