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

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

To find the first derivative, we will have to use the 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 .

##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##