How do you differentiate #f(x)=cos(3x)*(2/3tanx)# using the product rule?

Answer 1

#f'(x)=(2sec^2xcos3x-6tanxsin3x)/3#

According to the product rule:

#f'(x)=2/3tanxd/dx[cos3x]+cos3xd/dx[2/3tanx]#

Find each derivative separately. The first will require the chain rule:

#d/dx[cos3x]=-sin3xd/dx[3x]=-3sin3x#
#d/dx[2/3tanx]=2/3d/dx[tanx]=2/3sec^2x#

Plug these back in:

#f'(x)=2/3tanx(-3sin3x)+cos3x(2/3sec^2x)#
#f'(x)=2/3sec^2xcos3x-2tanxsin3x#
#f'(x)=(2sec^2xcos3x-6tanxsin3x)/3#
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Answer 2

To differentiate the function ( f(x) = \cos(3x) \cdot \frac{2}{3}\tan(x) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = \cos(3x) ) and ( v(x) = \frac{2}{3}\tan(x) ).
  2. Apply the product rule: ( f'(x) = u'(x)v(x) + u(x)v'(x) ).
  3. Differentiate each function separately: ( u'(x) = -3\sin(3x) ) and ( v'(x) = \frac{2}{3}\sec^2(x) ).
  4. Substitute the derivatives into the product rule formula.
  5. Simplify the expression to obtain the derivative of ( f(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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