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
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