What is the Second Derivative of xtanx?

[escryan]

Homework Statement

Given y=xtanx, find y'' (second derivative)

Homework Equations

Uh... I'm not even sure if I'm using the right one...
d/dx(tanx) = sec^2x

The Attempt at a Solution

y=xtanx
y'= (x)(sec^2(x)) + (tanx)(1)
y'= xsec^2(x) + tanx

y'' = [(x)(2sec^3(x)) + sec^2(x)(1)] + sec^2x
y'' = 2xsec^3(x) + sec^2(x) + sec^2(x)

...

The key answer says it is supposed to be (2cosx + 2xsinx)/(cos^3(x))

 

[rocomath]

you took the 2nd derivative wrong

$$y'=x\sec^{2}x+\tan x$$

$$y''=x\cdot2\sec^{2-1}x\cdot\sec x\tan x+\sec^{2}x+\sec^{2}x$$
$$=2x\sec^{2}x\tan x+2\sec^{2}x$$

What you did was increase the power rather than decreasing it.

In general, the derivative of secant raised to a power is ... $$\frac{d}{dx}(\sec^{n}x)=n\sec^{n}x\tan x$$

 

[escryan]

Oh wow, I'm an idiot..

So is the general rule d/dx(sec^n(x)) = nsec^n(x)tan(x) just a combination of the (x^n)' = nx^x-1 and d/dx(secx) = secxtanx?

What if I were to be givin d/dx(tan^n(x))... would the answer be like nsec^2(n-1)(x)? nsec^n(x)? ...

Thanks so much for your help, by the way :).

 

[rocomath]

No, don't forget the chain rule!

$$\frac{d}{dx}(\tan^{n}x)=n\tan^{n-1}x\sec^{2}x$$

 

[escryan]

The chain rule?
... I've never actually seen that before.
Haha, I guess that explains a few things! I haven't been taught that yet.

Guess I'll go read up on that, and thanks again for your help! I really appreciate it :).

 

[golfer747]

you took the 2nd derivative wrong

$$y'=x\sec^{2}x+\tan x$$

$$y''=x\cdot2\sec^{2-1}x\cdot\sec x\tan x+\sec^{2}x+\sec^{2}x$$
$$=2x\sec^{2}x\tan x+2\sec^{2}x$$

What you did was increase the power rather than decreasing it.

In general, the derivative of secant raised to a power is ... $$\frac{d}{dx}(\sec^{n}x)=n\sec^{n}x\tan x$$

i got the + sec^2x but how did you get the other + sec^2x