Answer:

General Formulas and Concepts:
<u>Pre-Algebra</u>
Order of Operations: BPEMDAS
- Brackets
- Parenthesis
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
<u>Algebra I</u>
<u>Calculus</u>
Derivatives
Derivative Notation
Derivative of a constant is 0
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
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 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)
ln Derivative: ![\displaystyle \frac{d}{dx} [lnu] = \frac{u'}{u}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5Blnu%5D%20%3D%20%5Cfrac%7Bu%27%7D%7Bu%7D)
Step-by-step explanation:
<u>Step 1: Define</u>
<u />
<u />
<u />
<u>Step 2: Differentiate</u>
- [Derivative] Chain Rule:
![\displaystyle y' = \frac{d}{dx}[ln(x^2 + 6)^{\frac{3}{2}}] \cdot \frac{d}{dx}[ln(x^2 + 6)] \cdot \frac{d}{dx}[x^2 + 6]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bln%28x%5E2%20%2B%206%29%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%5D%20%5Ccdot%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bln%28x%5E2%20%2B%206%29%5D%20%5Ccdot%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bx%5E2%20%2B%206%5D)
- [Derivative] Chain Rule [Basic Power Rule]:
![\displaystyle y' = \frac{3}{2}ln(x^2 + 6)^{\frac{3}{2} - 1} \cdot \frac{d}{dx}[ln(x^2 + 6)] \cdot \frac{d}{dx}[x^2 + 6]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%20%5Cfrac%7B3%7D%7B2%7Dln%28x%5E2%20%2B%206%29%5E%7B%5Cfrac%7B3%7D%7B2%7D%20-%201%7D%20%5Ccdot%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bln%28x%5E2%20%2B%206%29%5D%20%5Ccdot%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bx%5E2%20%2B%206%5D)
- [Derivative] Simplify:
![\displaystyle y' = \frac{3}{2}ln(x^2 + 6)^{\frac{1}{2}} \cdot \frac{d}{dx}[ln(x^2 + 6)] \cdot \frac{d}{dx}[x^2 + 6]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%20%5Cfrac%7B3%7D%7B2%7Dln%28x%5E2%20%2B%206%29%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%20%5Ccdot%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bln%28x%5E2%20%2B%206%29%5D%20%5Ccdot%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bx%5E2%20%2B%206%5D)
- [Derivative] ln Derivative:
![\displaystyle y' = \frac{3}{2}ln(x^2 + 6)^{\frac{1}{2}} \cdot \frac{1}{x^2 + 6} \cdot \frac{d}{dx}[x^2 + 6]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%20%5Cfrac%7B3%7D%7B2%7Dln%28x%5E2%20%2B%206%29%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%20%5Ccdot%20%5Cfrac%7B1%7D%7Bx%5E2%20%2B%206%7D%20%5Ccdot%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bx%5E2%20%2B%206%5D)
- [Derivative] Basic Power Rule:

- [Derivative] Simplify:

- [Derivative] Multiply:

- [Derivative] Multiply:

- [Derivative] Multiply:

- [Derivative] Multiply:

- [Derivative] Factor:

- [Derivative] Simplify:

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