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• Let f be a function such that f(-2) = 8 and f'(-2) = 4.
• Let h be the function h(x) = x3.
Evaluate
d f(x)
dx h(x)
at x = -2.
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Let f be a function such that f2 8 and f2 4 Let h be the function hx x3 Evaluate d fx dx hx at x 2 Show Calculator class=

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Answer:

-2

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

  1. Brackets
  2. Parenthesis
  3. Exponents
  4. Multiplication
  5. Division
  6. Addition
  7. Subtraction
  • Left to Right

Calculus

Derivatives

Basic Power Rule:

  • f(x) = cxⁿ
  • f’(x) = c·nxⁿ⁻¹

Quotient Rule: [tex]\frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}[/tex]

Step-by-step explanation:

Step 1: Define

[tex]\frac{d}{dx} [\frac{f(x)}{h(x)} ] \ at \ x = -2\\h(x) = x^3\\f(-2) = 8\\f'(-2) = 4[/tex]

Step 2: Differentiate

  1. Differentiate [Quotient Rule]:                    [tex]\frac{d}{dx} [\frac{f(x)}{h(x)} ] = \frac{f'(x)h(x) - f(x)h'(x)}{h(x)^2}[/tex]
  2. Differentiate h(x) [Basic Power]:                h'(x) = 3x²

Step 3: Evaluate

  1. Define differential:                     [tex]\frac{f'(x)x^3 + f(x)[3x^2]}{(x^3)^2}[/tex]
  2. Substitute in variables:             [tex]\frac{f'(-2)h(-2) + f(-2)h'(-2)}{h(-2)^2}[/tex]
  3. Substitute in variables:             [tex]\frac{4(-2)^3 - 8[3(-2)^2]}{[(-2)^3]^2}[/tex]
  4. Evaluate:                                    -2

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