How do you differentiate #f(x)=(x^2-2x+8)/sinx-1/x# using the quotient rule?
Let's start by rearranging the second term in f(x) to facilitate differentiation.
Next, let's apply the quotient rule to differentiate the first term.
By dividing the fraction and canceling the sin at the top and bottom, we can make this a little simpler.
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To differentiate the function ( f(x) = \frac{{x^2 - 2x + 8}}{{\sin x - \frac{1}{x}}} ) using the quotient rule, follow these steps:
- Let ( u(x) = x^2 - 2x + 8 ) and ( v(x) = \sin x - \frac{1}{x} ).
- Compute ( u'(x) ) and ( v'(x) ).
- Apply the quotient rule: ( \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} ).
- Substitute the values of ( u ), ( v ), ( u' ), and ( v' ) into the quotient rule formula.
- Simplify the expression to obtain the derivative of ( f(x) ).
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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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