How do you differentiate #f(x)=sin2x * cotx# using the product rule?

Answer 1

#2cos2x*cotx-csc^2x*sin2x#

The product rule: #d/(dx)[f(x)]=("derivative of the first term" * "the second term")+("derivative of the second term"*"the first term")#
#d/dx[cotx]=-csc^2x #
#d/(dx)[f(x)]= (d/dx[sin(2x)]*cotx)+(d/dx[cotx]*sin2x)# #=(2cos2x*cotx)+(-csc^2x*sin2x)# #=2cos2x*cotx-csc^2x*sin2x#
You could just stop there, or you could simplify the answer further by uniting all the angles as #x# using double angle formulas.
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Answer 2

To differentiate ( f(x) = \sin(2x) \cdot \cot(x) ) using the product rule, follow these steps:

  1. Identify the functions involved: ( u(x) = \sin(2x) ) and ( v(x) = \cot(x) ).
  2. Compute the derivatives of ( u(x) ) and ( v(x) ) with respect to ( x ): ( u'(x) = 2\cos(2x) ) and ( v'(x) = -\csc^2(x) ).
  3. Apply the product rule: ( f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x) ).
  4. Substitute the derivatives and functions back into the formula: [ f'(x) = (2\cos(2x)) \cdot (\cot(x)) + (\sin(2x)) \cdot (-\csc^2(x)) ].
  5. Simplify the expression if necessary.

So, the derivative of ( f(x) = \sin(2x) \cdot \cot(x) ) using the product rule is: [ f'(x) = 2\cos(2x) \cdot \cot(x) - \sin(2x) \cdot \csc^2(x) ].

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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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