How do you differentiate #f(x)=(x^2-2x+8)/sinx-1/x# using the quotient rule?

Answer 1

#f'(x) = (2x-2)/sinx - ((x^2-2x+8)cosx)/sin^2x - x^-2#

The quotient rule is #f'(x) = (v(du)/(dx) - u(dv)/(dx))/v^2#

Let's start by rearranging the second term in f(x) to facilitate differentiation.

#f(x) = (x^2 - 2x + 8)/sinx - x^-1#

Next, let's apply the quotient rule to differentiate the first term.

#u = x^2 - 2x + 8# #v = sinx#
#(du)/dx = 2x - 2#
#(dv)/dx = cosx#
#f'(x) = ((2x-2)sinx - (x^2-2x + 8)cosx)/(sinx)^2#

By dividing the fraction and canceling the sin at the top and bottom, we can make this a little simpler.

#f'(x) = (2x-2)/sinx - ((x^2-2x+8)cosx)/sin^2x#
Now we just put that back in and differentiate #x^-1#
#f'(x) = (2x-2)/sinx - ((x^2-2x+8)cosx)/sin^2x + x^-2#
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Answer 2

To differentiate the function ( f(x) = \frac{{x^2 - 2x + 8}}{{\sin x - \frac{1}{x}}} ) using the quotient rule, follow these steps:

  1. Let ( u(x) = x^2 - 2x + 8 ) and ( v(x) = \sin x - \frac{1}{x} ).
  2. Compute ( u'(x) ) and ( v'(x) ).
  3. Apply the quotient rule: ( \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} ).
  4. Substitute the values of ( u ), ( v ), ( u' ), and ( v' ) into the quotient 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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