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

Answers

Answer 1

Answer:

[tex]\displaystyle \frac{dy}{dx} = 2x(10x^3 - 30x - 7)[/tex]

General Formulas and Concepts:

Algebra I

Terms/CoefficientsFactoring/Expanding

Calculus

Differentiation

DerivativesDerivative Notation

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

Derivative Property [Addition/Subtraction]:                                                         [tex]\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)][/tex]

Basic Power Rule:

f(x) = cxⁿf’(x) = c·nxⁿ⁻¹

Explanation:

Step 1: Define

Identify

[tex]\displaystyle 4x^2(x^3 - 5x) - 7x^2[/tex]

Step 2: Differentiate

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

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

Unit: Differentiation

Book: College Calculus 10e


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