Finding the sixth derivative of sin(x²) at x=0

[me9900]

Homework Statement

Compute the sixth derivative of f(x)=sin(x^2) at x=0. Hint: Maclaurin polynomial may be helpful to you.

Homework Equations

The taylor expansion/maclaurin expansion for sin.

The Attempt at a Solution

Help.

 

[Mark44]

Hint: Maclaurin polynomial may be helpful to you.
Do you know the Maclaurin series for sin(x)?

 

[me9900]

yes but how does the x^2 alter the expansion?

 

[Mark44]

Replace each occurrence of x in the Maclaurin series expansion for sin(x) with x^2.

 

[me9900]

So the expansion would be:

x^2-x^6/3!+x^10/5! ?

 

[Mark44]

So the expansion would be:

x^2-x^6/3!+x^10/5! ?

That would be the first three nonzero terms, yes. Keep in mind there are quite a few terms whose coefficients are zero, so sin(x^2) = 0 + 0x + x^2 + 0x^3 + 0x^4 + 0x^5 - x^6/3! + ...

Regarding your original problem, keep in mind the general formula for the Maclaurin series, and how you get the coefficients of the various powers of x.

 

[me9900]

Thanks