- #1
makar
- 3
- 0
Homework Statement
how to find nth derivatives cos^12x and a-x/a+x
Finding the nth Derivatives of cos^12x & a-x/a+x
- Thread starter makar
- Start date
-
- Tags
- Derivatives
Click For Summary
SUMMARY
The discussion focuses on finding the nth derivatives of the functions cos^12(x) and (a-x)/(a+x). Participants suggest starting by calculating the first few derivatives of cos^12(x) to identify a pattern, similar to the nth derivative of sin(x). Two methods are proposed: using the expansion of cos^12(x) into a polynomial form and applying the Leibniz rule for derivatives. Both approaches aim to simplify the process of deriving these functions systematically.
PREREQUISITES- Understanding of basic differentiation rules
- Familiarity with trigonometric functions and their derivatives
- Knowledge of the Leibniz rule for derivatives
- Ability to recognize and analyze patterns in derivatives
- Study the Leibniz rule for derivatives in detail
- Practice finding derivatives of higher-order trigonometric functions
- Explore polynomial expansions of trigonometric functions
- Investigate the application of Taylor series in deriving trigonometric functions
Students studying calculus, mathematics educators, and anyone interested in advanced differentiation techniques for trigonometric functions.
how to find nth derivatives cos^12x and a-x/a+x
- #2
makar
- 3
- 0
help please..
- #3
LeonhardEu
- 33
- 3
Don't panic. Since you know derivation is quite easy to deduce them. Write the first two or three derivatives and see the pattern. I 'll give an example to understand:
(sinx)' = cosx = sin(x+pi/2)
(sinx)'' = (cosx)' = -sinx= sin(x+pi), (sinx)''' = -cosx = sin(x+3pi/2), (sinx)(4) = sinx = sin(x+2pi).
SO the n-nth derivative of sinx is sin(x + n*pi/2)
Yours are all the same way.
(sinx)' = cosx = sin(x+pi/2)
(sinx)'' = (cosx)' = -sinx= sin(x+pi), (sinx)''' = -cosx = sin(x+3pi/2), (sinx)(4) = sinx = sin(x+2pi).
SO the n-nth derivative of sinx is sin(x + n*pi/2)
Yours are all the same way.
- #4
tiny-tim
Science Advisor
Homework Helper
- 25,837
- 258
hi makar! welcome to pf! 
try it, and show us what you get
start with the first few derivatives of cos12x
(you may spot a pattern)
try it, and show us what you get
start with the first few derivatives of cos12x
(you may spot a pattern)
- #5
lurflurf
Homework Helper
- 2,459
- 159
There are two ways to go
1)use
=cos(x)^12=(792 cos(2 x)+495 cos(4 x)+220 cos(6 x)+66 cos(8 x)+12 cos(10 x)+cos(12 x)+462)/2048
2)use the Leibniz rule
http://en.wikipedia.org/wiki/Leibniz_rule_(generalized_product_rule)
For use (a-x)/(a+x)
(a-x)/(a+x)=(2 a)/(a+x)-1
1)use
=cos(x)^12=(792 cos(2 x)+495 cos(4 x)+220 cos(6 x)+66 cos(8 x)+12 cos(10 x)+cos(12 x)+462)/2048
2)use the Leibniz rule
http://en.wikipedia.org/wiki/Leibniz_rule_(generalized_product_rule)
For use (a-x)/(a+x)
(a-x)/(a+x)=(2 a)/(a+x)-1
- #6
makar
- 3
- 0
thank u everyone for helping me..
Similar threads
- · Replies 7 ·
- · Replies 10 ·
- · Replies 14 ·
- · Replies 8 ·
- · Replies 3 ·
- · Replies 27 ·