Question

Derivative?

Find the derivative of `\5xtan^2\(3x)`

Answer 1

Use the product rule and the chain rule.

Explanation:

`u = 5x`, so `u' = 5`

`v = tan^2(3x) = (tan(3x))^2` has derivative

`v' = 2tan(3x) * d/dx(tan(3x))`

`= 2tan(3x) [sec^2(3x) * d/dx(3x)]`

`= 2tan(3x) sec^2(3x) * 3`

`= 6tan(3x) sec^2(3x)`

`d/dx(uv) = u'v+uv'`

`= [5][tan^2(3x)] + [5x][6tan(3x) sec^2(3x)]`

`= 5tan^2(3x) + 30 x tan(3x) sec^2(3x)`

You may remove the common factors if you like.

`= 5tan(3x)[tan(3x)+6xsec^2(3x)]`