Use the Product Rule to find the derivative of : a.y=(3x-1)^3(2x+5) b. y=(3x-2)(2x^2+3)

Question

Use the Product Rule to find the derivative of :
a.y=(3x-1)^3(2x+5)

b. y=(3x-2)(2x^2+3)

in progress 0
Thu Thảo 5 years 2021-07-30T23:54:51+00:00 1 Answers 15 views 0

Answers ( )

  1. thuthuy
    0
    2021-07-30T23:56:16+00:00
    Reply

    Answer:

    see explanation

    Step-by-step explanation:

    Using the product rule

    Given

    y = f(x)g(x) , then

    \frac{dy}{dx} = f(x).g'(x) + g(x).f'(x) ← product rule

    (a)

    let

    f(x) = (3x – 1)³ , then using the chain rule

    f'(x) = 3(3x – 1)² × \frac{d}{dx}(3x – 1) = 3(3x – 1)² × 3 = 9(3x – 1)²

    let

    g(x) = 2x + 5 ⇒ g'(x) = 2

    Then

    \frac{dy}{dx} = (3x – 1)³ . 2 + (2x + 5) × 9(3x – 1)²

        = 2(3x – 1)³ +9(3x – 1)²(2x + 5) ← factor out (3x – 1)²

        = (3x – 1)²[ 2(3x – 1) + 9(2x + 5) ]

        = (3x – 1)² [ 6x – 2 + 18x + 45 ]

        = (3x – 1)²(24x + 43)

    ————————————————————–

    (b)

    let

    f(x) = 3x – 2 ⇒ f'(x) = 3

    let

    g(x) = 2x² + 3 ⇒ g'(x) = 4x

    Then

    \frac{dy}{dx} = (3x – 2).4x + (2x² + 3). 3 ← distribute parenthesis

        = 12x² – 8x + 6x² + 9

        = 12x² – 8x + 9

    Note

    It is usual to simplify after differentiating the expressions

Leave an answer

Browse

Giải phương trình 1 ẩn: x + 2 - 2(x + 1) = -x . Hỏi x = ? ( )