What is the second derivative of $f (x) = {sec}^{2} x$ $f(x)= sec^2x$ ?

Question

22068 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-01-28T20:58:30

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

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

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

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

What is the second derivative of f(x)= sec^2xf(x)=sec2xf(x)= sec^2x?