Question

I need to evaluate the derivative of the function and find the answer

Answer 1

Answer:

C.

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Multiplied Constant]:                                                           $\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)$

Derivative Property [Addition/Subtraction]:                                                         $\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]$  

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

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

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle f(t) = \frac{8t + 2}{4t - 1}$

Step 2: Differentiate

1. Derivative Rule [Quotient Rule]:                                                                  $\displaystyle f'(t) = \frac{(8t + 2)'(4t - 1) - (8t + 2)(4t - 1)'}{(4t - 1)^2}$
2. Basic Power Rule [Derivative Properties]:                                                  $\displaystyle f'(t) = \frac{8(4t - 1) - 4(8t + 2)}{(4t - 1)^2}$
3. Simplify:                                                                                                         $\displaystyle f'(t) = \frac{-16}{(4t - 1)^2}$

Step 3: Evaluate

1. Substitute in point [Derivative]:                                                                    $\displaystyle f'(3) = \frac{-16}{[4(3) - 1]^2}$
2. Simplify:                                                                                                         $\displaystyle f'(3) = \frac{-16}{121}$

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

Unit: Differentiation