How do you differentiate #f(x) = (e^x-4x)/(2x-e^x)# using the quotient rule?

Answer 1

The answer is #=(2e^x(1-x))/(2x-e^x)^2#

The rule of the quotient is

#(u/v)'=(u'v-uv')/(v^2)#

Here,

#u=e^x-4x#, #=>#, #u'=e^x-4#
#v=2x-e^x#, #=>#, #v'=2-e^x#

Consequently,

#f'(x)=((e^x-4)(2x-e^x)-(e^x-4x)(2-e^x))/(2x-e^x)^2#
#=(2xe^x-cancele^(2x)-cancel8x+4e^x-2e^x+cancele^(2x)+cancel8x-4xe^x)/(2x-e^x)^2#
#=(2e^x-2xe^x)/(2x-e^x)^2#
#=(2e^x(1-x))/(2x-e^x)^2#
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Answer 2

To differentiate ( f(x) = \frac{e^x - 4x}{2x - e^x} ) using the quotient rule, follow these steps:

  1. Let ( u = e^x - 4x ) and ( v = 2x - e^x ).
  2. Find the derivatives ( u' ) and ( v' ) with respect to ( x ).
  3. Apply the quotient rule: ( f'(x) = \frac{u'v - uv'}{v^2} ).
  4. Substitute the values of ( u ), ( v ), ( u' ), and ( v' ) into the formula.
  5. Simplify the expression to get 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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