Question

What is the second derivative of `f(x)= sec^2x`?

Answer 1

`f''(x)=4tan^2xsec^2x+2sec^4x`

Explanation:

To find the first derivative, we will have to use the chain rule on the second power.

Use the rule that `d/dx(u^2)=2u*u'`.

Thus, we see that

`f'(x)=2secx*d/dx(secx)`

`f'(x)=2secx*secxtanx`

`f'(x)=2sec^2xtanx`

To find the second derivative, we will have to use the product rule.

`f''(x)=2tanxd/dx(sec^2x)+2sec^2xd/dx(tanx)`

Note that we already know that `d/dx(sec^2x)=2sec^2xtanx` and that `d/dx(tanx)=sec^2x`.

This gives us

`f''(x)=2tanx(2sec^2xtanx)+2sec^2x(sec^2x)`

`f''(x)=4tan^2xsec^2x+2sec^4x`