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What is the derivative of #arctan (cos 2t)#?

Answer 1

Charles Anctil

# -2sin(2t)/(cos(2t)^2+1)#

The derivative of #tan^-1(x)# is #1/(x^2+1)#

when we substitute #cos(2t)# for x we get

#1/(cos(2t)^2+1)#

Then we apply the chain rule for cos(2t)

#1/(cos(2t)^2+1) * -2sin(2t)#

Our final answer is

# -2sin(2t)/(cos(2t)^2+1)#

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Answer 2

Charles Anctil

The derivative of arctan(cos(2t)) is -2sin(2t)/(1 + cos(2t))^2.

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Answer 3

Aurora Alvarado

The derivative of ( \arctan(\cos(2t)) ) with respect to ( t ) can be found using the chain rule and the derivative of the inverse tangent function. The derivative is:

[ \frac{d}{dt} \left( \arctan(\cos(2t)) \right) = -\frac{2\sin(2t)}{1 + \cos^2(2t)} ]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.