Find y' (a) by applying the product rule and (b) by multiplying the factors to produce a sum of simpler terms to differentiate. y = ( 4 − x 2 ) ( x 3 − 4 x + 4 )

Question

Find y&#039; (a) by applying the product rule and (b) by multiplying the factors to produce a sum of simpler terms to differentiate.  y  =  (  4  −      x          2        )  (      x          3        −  4  x  +  4  )

Malcolm Mcbride · Accepted Answer

y  =  (  4  −      x          2        )  (      x          3        −  4  x  +  4  )a.The product rule is       d          d      x        [  f  (  x  )  g  (  x  )  ]  =  f  (  x  )      g    ′    (  x  )  +  g  (  x  )      f    ′    (  x  )So,       y    ′    =  (  4  −      x          2        )  ∗  (  3      x          2        −  4  )  +  (      x          3        −  4  x  +  4  )  ∗  (  −  2  x  )  =  −  3      x          4        +  16      x          2        −  16  −  2      x          4        +  8      x          2        −  8  x  =  −  5      x          4        +  24      x          2        −  8  x  −  16b. They just want you to multiply   (  4  −      x          2        )  (      x          3        −  4  x  +  4  ) together and then differentiate.So,   y  =  (  4  −      x          2        )  (      x          3        −  4  x  +  4  )  =  −      x          5        +  8      x          3        −  4      x          2        −  16  x  +  16Now, differentiate using the power rule we get:      y    ′    =  −  5      x          4        +  24      x          2        −  8  x  −  16

Ebone6v · Accepted Answer

(a)      y    ′    =  −  2  x  (      x          3        −  4  x  +  4  )  +  (  4  −      x          2        )  (  3      x          2        −  4  )  =  −  2      x          4        +  8      x          2        −  8  x  +  12      x          2        −  16  −  3      x          4        +  4      x          2        =  −  5      x          4        +  24      x          2        −  8  x  −  16(b)  y  =  (  4  −      x          2        )  (      x          3        −  4  x  +  4  )  =  4      x          3        −  16  x  +  16  −      x          5        +  4      x          3        −  4      x          2        =  −      x          5        +  (  4  +  4  )      x          3        −  4      x          2        −  16  x  +  16  =  −      x          5        +  8      x          3        −  4      x          2        −  16  x  +  16      y    ′    =  −  5      x          4        +  24      x          3        −  8  x  −  16

Find y' (a) by applying the product rule and (b) by multiplying the factors to produce a sum of simpler terms to differentiate. y=(4-x^2)(x^3 -4x+4)

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