Answer:
g''(0) = 8
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
- Brackets
- Parenthesis
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
Algebra I
Calculus
Derivatives
Derivative Notation
Derivative Property [Addition/Subtraction]: [tex]\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)][/tex]
Derivative Rule [Product Rule]: [tex]\displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)[/tex]
Derivative: [tex]\displaystyle \frac{d}{dx} [e^u]=e^u \cdot u'[/tex]
Step-by-step explanation:
Step 1: Define
[Given] g(x) = eˣ · f(x)
- f(0) = 1
- f'(0) = 2
- f''(0) = 3
[Solve] g''(0)
Step 2: Differentiate
- [1st Derivative] Product Rule: g'(x) = eˣf(x) + eˣf'(x)
- [2nd Derivative] Product Rule [Derivative Property - Addition]: g''(x) = [eˣf(x) + eˣf'(x)] + [eˣf'(x) + eˣf''(x)]
- [2nd Derivative] Combine like terms [Addition]: g''(x) = eˣf(x) + 2eˣf'(x) + eˣf''(x)
Step 3: Evaluate
- [2nd Derivative] Substitute in variables: g''(0) = e⁰f(0) + 2e⁰f'(0) + e⁰f''(0)
- [2nd Derivative] Substitute in functions: g''(0) = e⁰(1) + 2e⁰(2) + e⁰(3)
- [2nd Derivative] Evaluate exponents: g''(0) = 1(1) + 2(1)(2) + 1(3)
- [2nd Derivative] Multiply: g''(0) = 1 + 4 + 3
- [2nd Derivative] Add: g''(0) = 8
Topic: AP Calculus AB/BC (Calculus I/II)
Unit: Derivatives
Book: College Calculus 10e
