[M1] Differentiation

Use Product Rule: d/dx (UV) =U dV/dx + V dU/dx

y=(3x+2)^5 x (6x-5)^3

Let U = (3x+2)^5 and V = (6x-5)^3
dy/dx =(3x+2)^5 d/dx[(6x-5)^3] + (6x-5)^3 d/dx[(3x+2)^5]
= [(3x+2)^5][(6)3(6x-5)^2] + [(6x-5)^3] [(3)5 (3x+2)^4]
= [(3x+2)^5][18(6x-5)^2] + [(6x-5)^3] [15(3x+2)^4]
= 18(3x+2)^5 (6x-5)^2 + 15(6x-5)^3 (3x+2)^4
= 18(3x+2)^5 (6x-5) ^2 + 15 (3x+2)^4 (6x-5)^3
= 3(3x+2)^4(6x-5)^2 [6(3x+2) + 5(6x-5)]
= 3(3x+2)^4(6x-5)^2 [18x+12 + 30x -25]
= 3(3x+2)^4(6x-5)^2 [48x-13]

dy/dx = 3(48x-13) (3x+2)^4 (6x-5)^2 (ANSWER)


Note:
If y = (3x+2)^5
Using chain rule
Let y = W^5
W = 3x + 2
dy/dW = 5W^4
dW/dx = 3
dy/dx = (dy/dW)(dW/dx)
= (5W^4) (3)
=15W^4
= 15(3x +2)^4

If y = (6x-5)^3
Using chain rule
Let y = Z^3
W = 6x-5
dy/dZ = 3Z^2
dZ/dx = 6
dy/dx = (dy/dZ)(dZ/dx)
= (3Z^2) (6)
=18Z^2
= 18(3x +2)^2