Deriving the Double Derivative of tan(x)

[_wolfgang_]

Homework Statement

the question asked to prove that the double derivaitve of y=tan(x) is...
2y(1+y^2)
eg. 2tanx(1+tan^2(x))

Homework Equations

The Attempt at a Solution

I was able to get the first derivative ( i think)
y=tan(x)
=(sin(x))/(cos(x))

dy/dx=(cos(x)cos(x)-(sin(x)(-sin(x))))/cos^2(x)
=(cos^2(x)+sin^2(x))/cos^2(x)
=1/cos^2(x)

from here i am not to sure how to get the second derivative...

 

[Borek]

So far so good, just differentiate cos^-2x.

 

[_wolfgang_]

So would i use the quotient rule to do that or something else?? like the chain rule??

 

[_wolfgang_]

ok iv done that i get
2sin(x)/cos^3(x)
now iv got the double derivative i can't see how to simplify it to get 2tanx(1+tan^2(x))

 

[Borek]

All I can tell is that

$$\frac {2 \sin x} {\cos^3 x} = \frac {2 \tan x } {\cos^2 x}$$

is a correct second derivative.

 

[_wolfgang_]

ok thanks for the help i should be able to get it from here