Answer:
a) 
b) 
General Formulas and Concepts:
<u>Pre-Calculus</u>
<u>Calculus</u>
Differentiation
- Derivatives
- Derivative Notation
Derivative Property [Multiplied Constant]: ![\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5Bcf%28x%29%5D%20%3D%20c%20%5Ccdot%20f%27%28x%29)
Derivative Property [Addition/Subtraction]:
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 [Quotient Rule]: ![\displaystyle \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=%5Cdisplaystyle%20%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)
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)
Trigonometric Differentiation
Logarithmic Differentiation
Step-by-step explanation:
a)
<u>Step 1: Define</u>
<em>Identify</em>

<u>Step 2: Differentiate</u>
- Logarithmic Differentiation [Chain Rule]:
![\displaystyle \frac{dy}{dx} = \frac{1}{\frac{1 - x}{\sqrt{1 + x^2}}} \cdot \frac{d}{dx}[\frac{1 - x}{\sqrt{1 + x^2}}]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%20%5Cfrac%7B1%7D%7B%5Cfrac%7B1%20-%20x%7D%7B%5Csqrt%7B1%20%2B%20x%5E2%7D%7D%7D%20%5Ccdot%20%5Cfrac%7Bd%7D%7Bdx%7D%5B%5Cfrac%7B1%20-%20x%7D%7B%5Csqrt%7B1%20%2B%20x%5E2%7D%7D%5D)
- Simplify:
![\displaystyle \frac{dy}{dx} = \frac{-\sqrt{x^2 + 1}}{x - 1} \cdot \frac{d}{dx}[\frac{1 - x}{\sqrt{1 + x^2}}]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%20%5Cfrac%7B-%5Csqrt%7Bx%5E2%20%2B%201%7D%7D%7Bx%20-%201%7D%20%5Ccdot%20%5Cfrac%7Bd%7D%7Bdx%7D%5B%5Cfrac%7B1%20-%20x%7D%7B%5Csqrt%7B1%20%2B%20x%5E2%7D%7D%5D)
- Quotient Rule:

- Basic Power Rule [Chain Rule]:

- Simplify:

- Simplify:

<u>Step 3: Find</u>
- Substitute in <em>x</em> = 0 [Derivative]:

- Evaluate:

b)
<u>Step 1: Define</u>
<em>Identify</em>

<u>Step 2: Differentiate</u>
- Logarithmic Differentiation [Chain Rule]:
![\displaystyle \frac{dy}{dx} = \frac{1}{\frac{1 + sinx}{1 - cosx}} \cdot \frac{d}{dx}[\frac{1 + sinx}{1 - cosx}]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%20%5Cfrac%7B1%7D%7B%5Cfrac%7B1%20%2B%20sinx%7D%7B1%20-%20cosx%7D%7D%20%5Ccdot%20%5Cfrac%7Bd%7D%7Bdx%7D%5B%5Cfrac%7B1%20%2B%20sinx%7D%7B1%20-%20cosx%7D%5D)
- Simplify:
![\displaystyle \frac{dy}{dx} = \frac{-[cos(x) - 1]}{sin(x) + 1} \cdot \frac{d}{dx}[\frac{1 + sinx}{1 - cosx}]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%20%5Cfrac%7B-%5Bcos%28x%29%20-%201%5D%7D%7Bsin%28x%29%20%2B%201%7D%20%5Ccdot%20%5Cfrac%7Bd%7D%7Bdx%7D%5B%5Cfrac%7B1%20%2B%20sinx%7D%7B1%20-%20cosx%7D%5D)
- Quotient Rule:
![\displaystyle \frac{dy}{dx} = \frac{-[cos(x) - 1]}{sin(x) + 1} \cdot \frac{(1 + sinx)'(1 - cosx) - (1 + sinx)(1 - cosx)'}{(1 - cosx)^2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%20%5Cfrac%7B-%5Bcos%28x%29%20-%201%5D%7D%7Bsin%28x%29%20%2B%201%7D%20%5Ccdot%20%5Cfrac%7B%281%20%2B%20sinx%29%27%281%20-%20cosx%29%20-%20%281%20%2B%20sinx%29%281%20-%20cosx%29%27%7D%7B%281%20-%20cosx%29%5E2%7D)
- Trigonometric Differentiation:
![\displaystyle \frac{dy}{dx} = \frac{-[cos(x) - 1]}{sin(x) + 1} \cdot \frac{cos(x)(1 - cosx) - sin(x)(1 + sinx)}{(1 - cosx)^2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%20%5Cfrac%7B-%5Bcos%28x%29%20-%201%5D%7D%7Bsin%28x%29%20%2B%201%7D%20%5Ccdot%20%5Cfrac%7Bcos%28x%29%281%20-%20cosx%29%20-%20sin%28x%29%281%20%2B%20sinx%29%7D%7B%281%20-%20cosx%29%5E2%7D)
- Simplify:
![\displaystyle \frac{dy}{dx} = \frac{-[cos(x) - sin(x) - 1]}{[sin(x) + 1][cos(x) - 1]}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%20%5Cfrac%7B-%5Bcos%28x%29%20-%20sin%28x%29%20-%201%5D%7D%7B%5Bsin%28x%29%20%2B%201%5D%5Bcos%28x%29%20-%201%5D%7D)
<u>Step 3: Find</u>
- Substitute in <em>x</em> = π/2 [Derivative]:
![\displaystyle \frac{dy}{dx} \bigg| \limit_{x = \frac{\pi}{2}} = \frac{-[cos(\frac{\pi}{2}) - sin(\frac{\pi}{2}) - 1]}{[sin(\frac{\pi}{2}) + 1][cos(\frac{\pi}{2}) - 1]}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bdy%7D%7Bdx%7D%20%5Cbigg%7C%20%5Climit_%7Bx%20%3D%20%5Cfrac%7B%5Cpi%7D%7B2%7D%7D%20%3D%20%5Cfrac%7B-%5Bcos%28%5Cfrac%7B%5Cpi%7D%7B2%7D%29%20-%20sin%28%5Cfrac%7B%5Cpi%7D%7B2%7D%29%20-%201%5D%7D%7B%5Bsin%28%5Cfrac%7B%5Cpi%7D%7B2%7D%29%20%2B%201%5D%5Bcos%28%5Cfrac%7B%5Cpi%7D%7B2%7D%29%20-%201%5D%7D)
- Evaluate [Unit Circle]:

Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
Book: College Calculus 10e