Question

Differentiate the following
4x²(x³ - 5x) - 7x²?​

Answer 1

`\displaystyle (dy)/(dx) = 2x(10x^3 - 30x - 7)`

General Formulas and Concepts:

Algebra I

• Terms/Coefficients
• Factoring/Expanding

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

Step-by-step explanation:

Step 1: Define

Identify

`\displaystyle 4x^2(x^3 - 5x) - 7x^2`

Step 2: Differentiate

1. Expand:
   `\displaystyle 4x^5 - 20x^3 - 7x^2`
2. Derivative Property [Addition/Subtraction]:
   `\displaystyle (dy)/(dx) = (d)/(dx)[4x^5] - (d)/(dx)[20x^3] - (d)/(dx)[7x^2]`
3. Derivative Property [Multiplied Constant]:
   `\displaystyle (dy)/(dx) = 4(d)/(dx)[x^5] - 20(d)/(dx)[x^3] - 7(d)/(dx)[x^2]`
4. Basic Power Rule:
   `\displaystyle (dy)/(dx) = 4(5x^4) - 20(3x^2) - 7(2x)`
5. Simplify:
   `\displaystyle (dy)/(dx) = 20x^4 - 60x^2 - 14x`
6. Factor:
   `\displaystyle (dy)/(dx) = 2x(10x^3 - 30x - 7)`

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

Unit: Differentiation

Book: College Calculus 10e