Question

The derivative (4x+1)^2

Answer 1

`\displaystyle (d)/(dx)[(4x + 1)^2] = 8(4x + 1)`

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Multiplied Constant]:
`\displaystyle (d)/(dx) [cf(x)] = c \cdot f'(x)`

Derivative Property [Addition/Subtraction]:
`\displaystyle (d)/(dx)[f(x) + g(x)] = (d)/(dx)[f(x)] + (d)/(dx)[g(x)]`

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

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

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 (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