Question

How do you calculate the fourth derivative of `f(x)=2x^4+3sin2x+(2x+1)^4`?

Answer 1

`y'''' = 432 + 48sin(2x)`

Explanation:

Application of the chain rule makes this problem easy, though it still requires some legwork to get to the answer:
`y = 2x^4+3sin(2x)+(2x+1)^4`
`y' = 8x^3 +6cos(2x)+8(2x+1)^3`
`y'' = 24x^2 -12sin(2x)+48(2x+1)^2`
`y''' = 48x - 24cos(2x)+192(2x+1)`
`=432x - 24cos(2x) + 192`

Note that the last step allowed us to substantially simplify the equation, making the final derivative much easier:
`y'''' = 432 + 48sin(2x)`