Answer:
![\displaystyle y' = -100x^4(2x^5 - 1)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%20-100x%5E4%282x%5E5%20-%201%29)
General Formulas and Concepts:
<u>Pre-Algebra</u>
<u>Algebra I</u>
- Terms/Coefficients
- Factoring
- Functions
- Function Notation
<u>Calculus</u>
Derivatives
Derivative Notation
Derivative Property [Multiplied Constant]:
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)
Step-by-step explanation:
<u>Step 1: Define</u>
<em>Identify</em>
<em />
<em />
<em />
<u>Step 2: Differentiate</u>
- Product Rule:
![\displaystyle y' = \frac{d}{dx}[(-4x^5 + 4)]5x^5 + (-4x^5 + 4)\frac{d}{dx}[5x^5]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%20%5Cfrac%7Bd%7D%7Bdx%7D%5B%28-4x%5E5%20%2B%204%29%5D5x%5E5%20%2B%20%28-4x%5E5%20%2B%204%29%5Cfrac%7Bd%7D%7Bdx%7D%5B5x%5E5%5D)
- Basic Power Rule [Derivative Property - Addition/Subtraction]:
![\displaystyle y' = (5 \cdot -4x^{5 - 1} + 0)5x^5 + (-4x^5 + 4)(5 \cdot 5x^{5 - 1})](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%20%285%20%5Ccdot%20-4x%5E%7B5%20-%201%7D%20%2B%200%295x%5E5%20%2B%20%28-4x%5E5%20%2B%204%29%285%20%5Ccdot%205x%5E%7B5%20-%201%7D%29)
- Simplify:
![\displaystyle y' = (-20x^4)5x^5 + (-4x^5 + 4)(25x^4)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%20%28-20x%5E4%295x%5E5%20%2B%20%28-4x%5E5%20%2B%204%29%2825x%5E4%29)
- Factor:
![\displaystyle y' = 5x^4 \bigg[ (-20x^4)x + (-4x^5 + 4)5 \bigg]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%205x%5E4%20%5Cbigg%5B%20%28-20x%5E4%29x%20%2B%20%28-4x%5E5%20%2B%204%295%20%5Cbigg%5D)
- [Distributive Property] Distributive parenthesis:
![\displaystyle y' = 5x^4 \bigg[ -20x^5 - 20x^5 + 20 \bigg]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%205x%5E4%20%5Cbigg%5B%20-20x%5E5%20-%2020x%5E5%20%2B%2020%20%5Cbigg%5D)
- Combine like terms:
![\displaystyle y' = 5x^4(-40x^5 + 20)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%205x%5E4%28-40x%5E5%20%2B%2020%29)
- Factor:
![\displaystyle y' = -100x^4(2x^5 - 1)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%20-100x%5E4%282x%5E5%20-%201%29)
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Derivatives
Book: College Calculus 10e