How do you differentiate #f(x)=cot(3x) # using the chain rule?

Answer 1

#frac{du}{dx} = -3csc^2(3x)#

First, I assume you know that the derivative of #cotx# is #-csc^2x#.
We substitute #u=3x#.

Therefore,

#frac{du}{dx} = 3#.

Now we use the chain rule.

#frac{d}{dx}(cot(3x)) = frac{d}{dx}(cot(u))#
#= frac{d}{du}(cot(u))*frac{du}{dx}#
# = -csc^2(u) * 3#
#= -3csc^2(3x)#
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Answer 2

To differentiate ( f(x) = \cot(3x) ) using the chain rule:

  1. Identify the outer function ( \cot(3x) ) and the inner function ( 3x ).
  2. Differentiate the outer function with respect to its inner function: ( \frac{d}{dx} \cot(u) = -\csc^2(u) \cdot \frac{du}{dx} ), where ( u = 3x ).
  3. Differentiate the inner function ( 3x ) with respect to ( x ): ( \frac{d}{dx}(3x) = 3 ).
  4. Substitute the results from steps 2 and 3 into the chain rule formula to get the final result: ( \frac{d}{dx} \cot(3x) = -3\csc^2(3x) ).
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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.

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.

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.

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.

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