Question

Find the indicated derivative.

Answer 1

Answer:

$\displaystyle \frac{d}{dt} \bigg( \frac{(9t - 8)^6}{t + 2} \bigg) = \frac{(9t - 8)^5(45t + 116)}{(t + 2)^2}$

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

Derivative Rule [Chain Rule]:                                                                                 $\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)$

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle \frac{d}{dt} \bigg( \frac{(9t - 8)^6}{t + 2} \bigg)$

Step 2: Differentiate

1. Derivative Rule [Quotient Rule]:                                                                   $\displaystyle \frac{d}{dt} \bigg( \frac{(9t - 8)^6}{t + 2} \bigg) = \frac{[(9t - 8)^6]'(t + 2) - (9t - 8)^6(t + 2)'}{(t + 2)^2}$
2. Basic Power Rule [Chain Rule, Multiplied Constant]:                                 $\displaystyle \frac{d}{dt} \bigg( \frac{(9t - 8)^6}{t + 2} \bigg) = \frac{6(9t - 8)^5(9t - 8)'(t + 2) - (9t - 8)^6}{(t + 2)^2}$
3. Basic Power Rule [Multiplied Constant, Addition/Subtraction]:                 $\displaystyle \frac{d}{dt} \bigg( \frac{(9t - 8)^6}{t + 2} \bigg) = \frac{54(9t - 8)^5(t + 2) - (9t - 8)^6}{(t + 2)^2}$
4. Factor:                                                                                                           $\displaystyle \frac{d}{dt} \bigg( \frac{(9t - 8)^6}{t + 2} \bigg) = \frac{(9t - 8)^5 \big[ 54(t + 2) - (9t - 8) \big] }{(t + 2)^2}$
5. Expand:                                                                                                         $\displaystyle \frac{d}{dt} \bigg( \frac{(9t - 8)^6}{t + 2} \bigg) = \frac{(9t - 8)^5 \big[ 54t + 108 - 9t + 8 \big] }{(t + 2)^2}$
6. Combine like terms:                                                                                     $\displaystyle \frac{d}{dt} \bigg( \frac{(9t - 8)^6}{t + 2} \bigg) = \frac{(9t - 8)^5(45t + 116)}{(t + 2)^2}$

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

Unit: Differentiation