- #1
makar
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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
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- Derivatives
In summary, to find the nth derivative of cos^12x and (a-x)/(a+x), you can use a pattern or the Leibniz rule. The first few derivatives of cos^12x are 792 cos(2x) + 495 cos(4x) + 220 cos(6x) + 66 cos(8x) + 12 cos(10x) + cos(12x) + 462/2048. For (a-x)/(a+x), you can use (2a)/(a+x) - 1.
how to find nth derivatives cos^12x and a-x/a+x
- #2
makar
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help please..
- #3
LeonhardEu
- 33
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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
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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
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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..
1. What is the formula for finding the nth derivative of cos^12x?
The formula for finding the nth derivative of cos^12x is (-1)^n * (12)^n * cos^(12-n)x. This can be derived using the power rule for derivatives and the chain rule.
2. Can you explain the steps to find the nth derivative of cos^12x?
To find the nth derivative of cos^12x, you can follow these steps:
1. Rewrite cos^12x as (cosx)^12
2. Apply the power rule for derivatives, which is d/dx(u^n) = n * u^(n-1) * du/dx
3. Use the chain rule to find the derivative of cosx, which is -sinx
4. Simplify the expression and replace n with the desired derivative number.
5. Multiply the final expression by (-1)^n to account for the alternating signs.
3. How does the value of n affect the nth derivative of cos^12x?
The value of n affects the nth derivative of cos^12x by changing the number of times the derivative will be taken. For example, if n=1, the first derivative of cos^12x will be taken. If n=2, the second derivative will be taken, and so on. The value of n also affects the final expression by determining the power of the cosine function that will remain after taking the derivative.
4. What is the general formula for finding the nth derivative of a-x/a+x?
The general formula for finding the nth derivative of a-x/a+x is (-1)^n * n! * (a-x)^(-n-1) * (a+x)^(-1). This can be derived using the quotient rule and the power rule for derivatives.
5. Can the nth derivative of a-x/a+x be simplified further?
No, the nth derivative of a-x/a+x cannot be simplified further. The final expression obtained using the quotient rule and the power rule for derivatives is the most simplified form. However, you can expand the expression and rewrite it in a different way if needed.
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