Question

Derivative of (x^7+2x-3)^3

Answer 1

`\displaystyle (dy)/(dx) = 3(7x^6 + 2)(x^7 + 2x - 3)^2`

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 = (x^7 + 2x - 3)^3`

Step 2: Differentiate

1. Basic Power Rule [Derivative Rule - Chain Rule]:
   `\displaystyle y' = 3(x^7 + 2x - 3)^2(x^7 + 2x - 3)'`
2. Basic Power Rule [Derivative Properties]:
   `\displaystyle y' = 3(7x^6 + 2)(x^7 + 2x - 3)^2`

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

Unit: Differentiation