Answer:
A. G'(5) = 20
B. G'(5) = -1
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
- Brackets
- Parenthesis
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
Algebra I
- Functions
- Function Notation
Calculus
Derivatives
Derivative Notation
Derivative Rule [Product Rule]: [tex]\displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)[/tex]
Derivative Rule [Quotient Rule]: [tex]\displaystyle \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
[Given] F(5) = 4, F'(5) = 4, H(5) = 2, H'(5) = 3
[Given] A. G(z) = F(z) · H(z)
[Given] B. G(w) = F(w) / H(w)
[Find] G'(5)
Step 2: Differentiate
A. G(z) = F(z) · H(z)
- [Derivative] Product Rule: G'(z) = F'(z)H(z) + F(z)H'(z)
B. G(w) = F(w) / H(w)
- [Derivative] Quotient Rule: G'(w) = [F'(w)H(w) - F(w)H'(w)] / H²(w)
Step 3: Evaluate
A. G'(5)
- Substitute in x [Function]: G'(5) = F'(5)H(5) + F(5)H'(5)
- Substitute in function values: G'(5) = 4(2) + 4(3)
- Multiply: G'(5) = 8 + 12
- Add: G'(5) = 20
B. G'(5)
- Substitute in x [Function]: G'(5) = [F'(5)H(5) - F(5)H'(5)] / H²(5)
- Substitute in function values: G'(5) = [4(2) - 4(3)] / 2²
- Exponents: G'(5) = [4(2) - 4(3)] / 4
- [Brackets] Multiply: G'(5) = [8 - 12] / 4
- [Brackets] Subtract: G'(5) = -4 / 4
- Divide: G'(5) = -1
Topic: AP Calculus AB/BC (Calculus I/II)
Unit: Derivatives
Book: College Calculus 10e
