<span>The mixed number is</span><span> = 2 <span>2/5.</span></span>
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
You use Pythagorean theorem to solve this problem.a^2+b^2=c^2
A.53*53=45*45+28*28
2809=2025+784
2809=2809
B.85*85=13*13+84*84
7725=169+7056
7725=7725
C.85*85=36*36+77*77
7725=1296+5929
7725=7725
D.65*65=16*16+61*61
4225=256+3721
4225=3977
The answer is D
You just have to look at the table that give you it was angle 31 so then you would go to angle and go across the line to where it says cos then you right the number which would be 0.8572