Finding 15th derivative shortcut

[sophiagrace]

Let

Compute the derivatives


I began constructing the series for x^2 cos x^3, but I am having trouble finding a shortcut to take 15 derivatives so I don't have to do them all out.
Hopefully somebody can help me. Thanks!

 

[Dr.Peterson]

Please show your actual work, particularly several terms of the series you found. Did you integrate it? Did you think about how it relates to the derivatives?

 

[sophiagrace]

Here is what I have so far. I built the series for Cos (x^3) and then integrated that series. I tried to find the pattern and apply it to find the 15th derivative, but it seems like I skipped a step...

 

[Dr.Peterson]

I think you're confusing the 15th derivative with the 15th term!

What term of a Maclaurin series contains the 15th derivative of the function? Write out what that term is.

Then compare that term to one you have already written! (Well, you haven't actually written it yet, because you didn't distribute.)

 

[pka]

Let View attachment 28827

Compute the derivatives View attachment 28828


I began constructing the series for x^2 cos x^3, but I am having trouble finding a shortcut to take 15 derivatives so I don't have to do them all out.
Hopefully somebody can help me. Thanks!

Good for you for correcting the notation above to [imath]D_x\left[\displaystyle{x^2}\int_0^x {\cos \left( {{t^3}} \right)dt}\right] [/imath]
What is the first derivative?

 

[sophiagrace]

I think you're confusing the 15th derivative with the 15th term!

What term of a Maclaurin series contains the 15th derivative of the function? Write out what that term is.

Then compare that term to one you have already written! (Well, you haven't actually written it yet, because you didn't distribute.)

Yes, I think I was confusing the two. The 15th derivative would be the 16th term, correct? If that's the case, I'm still getting the same term I have shown at the bottom of my work...

 

[Steven G]

View attachment 28835
Here is what I have so far. I built the series for Cos (x^3) and then integrated that series. I tried to find the pattern and apply it to find the 15th derivative, but it seems like I skipped a step...

Do you really think that the series for cos x ends, that is it is not an infinite series???

 

[Dr.Peterson]

Yes, I think I was confusing the two. The 15th derivative would be the 16th term, correct? If that's the case, I'm still getting the same term I have shown at the bottom of my work...

No, it's the term containing x^15! It's the 16th term only if you show all terms, including those that are zero!

The first few terms in that sense are 0 + 0x + 0x^2 + 1x^3 + ... .