How do you differentiate #f(x) = x^3/(xsinx-3)# using the quotient rule?
just follow the rule
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To differentiate ( f(x) = \frac{x^3}{x\sin(x) - 3} ) using the quotient rule, follow these steps:
- Identify ( u(x) = x^3 ) and ( v(x) = x\sin(x) - 3 ).
- Compute ( u'(x) ) and ( v'(x) ).
- ( u'(x) = 3x^2 )
- ( v'(x) = \sin(x) + x\cos(x) )
- Apply the quotient rule: [ f'(x) = \frac{u'(x)v(x) - v'(x)u(x)}{[v(x)]^2} ]
- Substitute the values: [ f'(x) = \frac{(3x^2)(x\sin(x) - 3) - (\sin(x) + x\cos(x))(x^3)}{(x\sin(x) - 3)^2} ]
- Simplify the expression if necessary.
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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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