Calculating the nth Derivative of Cos(X)

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In summary, the nth derivative of cos(x) follows a pattern where it repeats every fourth derivative and can be one of four choices depending on the remainder of n divided by 4. These choices are -sin(x), -cos(x), sin(x), or cos(x). This information may be helpful when working with Taylor Series.
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trumpetplaya1687
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nth derivative!

I have a quick question...
what is the nth derivative of cos(X)?? I have written out the derivatives up to the 4th one...it being cos(X) as well, but i don't see the pattern for the nth because it continues to change between sin and cos. I'm doing this for Taylor Series. Thanks for any help.
 
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trumpetplaya1687 said:
I have a quick question...
what is the nth derivative of cos(X)?? I have written out the derivatives up to the 4th one...it being cos(X) as well, but i don't see the pattern for the nth because it continues to change between sin and cos. I'm doing this for Taylor Series. Thanks for any help.

Well you have f(x) = cos(x) right

so

f'(x) = -sin(x) 1st derivative
f"(x) = -cos(x) 2nd derivative
f"'(x) = sin(x) 3rd derivative
f""(x) = cos(x) 4th derivative.
and it would repeat after this right...

see the pattern for a given n the nth derivative of cosine x can only be one of those 4 choices right.

so if n/4 has a remainder of 1 the nth derivative is -sin(x)
if n/4 has a remainder of 2 the nth derivative is -cos(x)
if n/4 has a remainder of 3 the nth derivative is sin(x)
if n/4 has a remainder of 0 ( n is divisible by 4) then the nth derivative is cos(x).

Does this help at all?
 

FAQ: Calculating the nth Derivative of Cos(X)

What is the formula for calculating the nth derivative of cos(x)?

The formula for calculating the nth derivative of cos(x) is (-1)^n * cos(x + nπ/2), where n is the order of the derivative.

How do you find the nth derivative of cos(x) using the power rule?

To find the nth derivative of cos(x) using the power rule, you must first rewrite cos(x) as (1/2)(e^(ix) + e^(-ix)). Then, you can use the power rule to find the derivative of each term and simplify to get the final formula.

What is the pattern for the derivatives of cos(x)?

The pattern for the derivatives of cos(x) is that the sign of the coefficient alternates between positive and negative, and the argument inside the cosine function increases by π/2 for each successive derivative.

How do you use the chain rule to find the nth derivative of cos(x)?

To use the chain rule to find the nth derivative of cos(x), you must first rewrite cos(x) as cos(u), where u = x. Then, you can use the chain rule to find the derivative of each term, and substitute back in for u to get the final formula.

What is the relationship between the nth derivative of cos(x) and the Taylor series for cos(x)?

The nth derivative of cos(x) is equal to the nth term in the Taylor series for cos(x). This means that by finding the nth derivative of cos(x), you are essentially finding the coefficient of the nth term in the Taylor series expansion of cos(x).

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