Question

What is the second derivative of `f(x)=tan(3x)`?

Answer 1

`f''(x)=18sec^2(3x)tan(3x)`

Explanation:

We will use the chain rule, together with the derivatives:

• `d/dx tan(x) = sec^2(x)`

• `d/dx 3x = 3`

• `d/dx x^2 = 2x`

• `d/dx sec(x) = sec(x)tan(x)`

First Derivative:
`f'(x) = d/dx tan(3x)`

`= sec^2(3x)(d/dx3x)`

`= 3sec^2(3x)`

Second Derivative:

`f''(x) = d/dx(f'(x))`

`= d/dx(3sec^2(3x))`

`= 3d/dx(sec^2(3x))`

`= 3(2sec(3x))(d/dxsec(3x))`

`=6sec(3x)(sec(3x)tan(3x))(d/dx3x)`

`=18sec^2(3x)tan(3x)`