Differentiate y = x(x+3)^4

 To differentiate this function we use the product rule. In the product rule we, leave the first alone, differentiate the second, and leave the second alone, differentiate the first.                                                                                                                  Say y = U * V (U and V are both functions of x)                                                                                                                      Then in general we have y' = U dV/dx + V dU/dx For this example we have; y' = x * d/dx (x+3)4 +  (x+3)4 d/dx x                      y' = x * 4 * (x+3)3 * 1 +  (x+3)4  * 1                                                                                                                                        When differentiating bracketed term we start differentiating outside the brackets and work our way in, therefore initially treating the brackets like a single term then accounting for the terms inside the brackets. To simplify the final expression we now take out the common factor.                                                                                                                                                       y' =  (x+3)3 [4x + (x+3)]                                                                                                                                                         Therefore, y' = (x+3)3 (5x+3)

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