Answer:
-2
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
- Brackets
- Parenthesis
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
Calculus
Derivatives
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Quotient Rule:
![(d)/(dx) [(f(x))/(g(x)) ]=(g(x)f'(x)-g'(x)f(x))/(g^2(x))](https://img.qammunity.org/2021/formulas/mathematics/college/it8u9heerf1b4gbeitcwifl1w9ck0iqkzk.png)
Explanation:
Step 1: Define
![(d)/(dx) [(f(x))/(h(x)) ] \ at \ x = -2\\h(x) = x^3\\f(-2) = 8\\f'(-2) = 4](https://img.qammunity.org/2021/formulas/mathematics/high-school/lbc18kohqrpr4ihxhzgn19iue14q3zfl1v.png)
Step 2: Differentiate
- Differentiate [Quotient Rule]:
![(d)/(dx) [(f(x))/(h(x)) ] = (f'(x)h(x) - f(x)h'(x))/(h(x)^2)](https://img.qammunity.org/2021/formulas/mathematics/high-school/qefj3190cp9uldcpfhs5fh28rp9t9mynvv.png)
- Differentiate h(x) [Basic Power]: h'(x) = 3x²
Step 3: Evaluate
- Define differential:
![(f'(x)x^3 + f(x)[3x^2])/((x^3)^2)](https://img.qammunity.org/2021/formulas/mathematics/high-school/7reublbfb5k69qt9bvvrvt297qyfa0ixzv.png)
- Substitute in variables:

- Substitute in variables:
![(4(-2)^3 - 8[3(-2)^2])/([(-2)^3]^2)](https://img.qammunity.org/2021/formulas/mathematics/high-school/33ozyx4k7ojgdm8zxcwdwhjgdrxa1uvhte.png)
- Evaluate: -2