Question

The derivative (4x+1)^2

Answer 1

Answer:

$\displaystyle \frac{d}{dx}[(4x + 1)^2] = 8(4x + 1)$

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

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle y = (4x + 1)^2$

Step 2: Differentiate

1. Basic Power Rule [Derivative Rule - Chain Rule]:                                       $\displaystyle y' = 2(4x + 1) \cdot \frac{d}{dx}[4x + 1]$
2. Basic Power Rule [Addition/Subtraction, Multiplied Constant]:                 $\displaystyle y' = 2(4x + 1) \cdot 4$
3. Simplify:                                                                                                         $\displaystyle y' = 8(4x + 1)$

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

Unit: Differentiation