Question

Please, someone, help very hard

Answer 1

A. G'(5) = 20

B. G'(5) = -1

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

Algebra I

• Functions
• Function Notation

Calculus

Derivatives

Derivative Notation

Derivative Rule [Product Rule]:
`\displaystyle (d)/(dx) [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)`

Derivative Rule [Quotient Rule]:
`\displaystyle (d)/(dx) [(f(x))/(g(x)) ]=(g(x)f'(x)-g'(x)f(x))/(g^2(x))`

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)

1. [Derivative] Product Rule: G'(z) = F'(z)H(z) + F(z)H'(z)

B. G(w) = F(w) / H(w)

1. [Derivative] Quotient Rule: G'(w) = [F'(w)H(w) - F(w)H'(w)] / H²(w)

Step 3: Evaluate

A. G'(5)

1. Substitute in x [Function]: G'(5) = F'(5)H(5) + F(5)H'(5)
2. Substitute in function values: G'(5) = 4(2) + 4(3)
3. Multiply: G'(5) = 8 + 12
4. Add: G'(5) = 20

B. G'(5)

1. Substitute in x [Function]: G'(5) = [F'(5)H(5) - F(5)H'(5)] / H²(5)
2. Substitute in function values: G'(5) = [4(2) - 4(3)] / 2²
3. Exponents: G'(5) = [4(2) - 4(3)] / 4
4. [Brackets] Multiply: G'(5) = [8 - 12] / 4
5. [Brackets] Subtract: G'(5) = -4 / 4
6. Divide: G'(5) = -1

Topic: AP Calculus AB/BC (Calculus I/II)

Unit: Derivatives

Book: College Calculus 10e