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# 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

### 1 Answer

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- 9 years agoFavorite Answer
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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