Calculus question involving the chain rule?

Use the chain rule, in leibniz notation, to find dy/dx at the given value of x

y=u(u^2 + 3 ) ^3 , u=(x+3)^2, x= - 2

Please help me im really confused

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  • 9 years ago
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    y = u(u² + 3)³, where u = (x + 3)², and you're given x = -2.

    Start off by deriving y and u separately:

    y' = u'(u² + 3)³ + 3u(u² + 3)²(2u) ... ... ...product rule and power/chain rule

    u' = 2(x + 3)(1) = 2x + 6

    Substitute u' back into y':

    y' = (2x + 6)(((x + 3)²)² + 3)³ + 3(x + 3)²(((x + 3)²)² + 3)²(2(x + 3)²)

    y' = (2x + 6)((x + 3)⁴ + 3)³ + 6(x + 3)⁴((x + 3)⁴ + 3)²

    y' = dy/dx, so now find dy/dx when x = -2:

    dy/dx = (2(-2) + 6)(((-2) + 3)⁴ + 3)³ + 6((-2) + 3)⁴(((-2) + 3)⁴ + 3)²

    = (2)(4)³ + 6(1)⁴(4)²

    = 128 + 96

    = 224

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