What is the Second Derivative of xtanx?

In summary: The key answer says it is supposed to be (2cosx + 2xsinx)/(cos^3(x))you took the 2nd derivative wrong.
  • #1
escryan
13
0

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))
 
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  • #2
you took the 2nd derivative wrong

[tex]y'=x\sec^{2}x+\tan x[/tex]

[tex]y''=x\cdot2\sec^{2-1}x\cdot\sec x\tan x+\sec^{2}x+\sec^{2}x[/tex]
[tex]=2x\sec^{2}x\tan x+2\sec^{2}x[/tex]

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

In general, the derivative of secant raised to a power is ... [tex]\frac{d}{dx}(\sec^{n}x)=n\sec^{n}x\tan x[/tex]
 
Last edited:
  • #3
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 :).
 
  • #4
No, don't forget the chain rule!

[tex]\frac{d}{dx}(\tan^{n}x)=n\tan^{n-1}x\sec^{2}x[/tex]
 
  • #5
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 :).
 
  • #6
rocomath said:
you took the 2nd derivative wrong

[tex]y'=x\sec^{2}x+\tan x[/tex]

[tex]y''=x\cdot2\sec^{2-1}x\cdot\sec x\tan x+\sec^{2}x+\sec^{2}x[/tex]
[tex]=2x\sec^{2}x\tan x+2\sec^{2}x[/tex]

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

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

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

1. What is the definition of the second derivative of xtanx?

The second derivative of xtanx is the derivative of the first derivative of xtanx. In other words, it is the rate of change of the slope of the tangent line to the curve of xtanx.

2. How do you find the second derivative of xtanx?

To find the second derivative of xtanx, you can use the power rule or the product rule. First, take the derivative of xtanx using the power rule or the product rule. Then, take the derivative of that result to find the second derivative.

3. What is the significance of the second derivative of xtanx?

The second derivative of xtanx can tell us about the concavity of the curve of xtanx. If the second derivative is positive, the curve is concave up, and if it is negative, the curve is concave down. It can also help us find the points of inflection of the curve.

4. Can the second derivative of xtanx be undefined?

Yes, the second derivative of xtanx can be undefined at certain points. This can happen if the first derivative of xtanx is discontinuous or if the tangent line is vertical at a certain point. In these cases, the second derivative is undefined because it does not exist.

5. How can the second derivative of xtanx be applied in real-life situations?

The second derivative of xtanx can be applied in real-life situations such as physics and engineering. For example, it can be used to analyze the acceleration of an object moving in a curved path or to calculate the maximum and minimum values of a function. It can also be used in economics to analyze the concavity of a production or cost curve.

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