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How do you differentiate #f(x)=(x-xsinx+1)/(x^2-3x)# using the quotient rule?

Answer 1

Ezekiel Alexander

See the answer below:

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Answer 2

Elijah Abram

To differentiate ( f(x) = \frac{x - x \cdot \sin(x) + 1}{x^2 - 3x} ) using the quotient rule:

Identify ( u(x) ) and ( v(x) ). ( u(x) = x - x \cdot \sin(x) + 1 ) ( v(x) = x^2 - 3x )
Apply the quotient rule: ( f'(x) = \frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{(v(x))^2} )
Find the derivatives of ( u(x) ) and ( v(x) ): ( u'(x) = 1 - (\sin(x) + x \cdot \cos(x)) ) ( v'(x) = 2x - 3 )
Substitute the derivatives and original functions into the quotient rule formula: ( f'(x) = \frac{(x^2 - 3x)(1 - (\sin(x) + x \cdot \cos(x))) - (x - x \cdot \sin(x) + 1)(2x - 3)}{(x^2 - 3x)^2} )
Simplify the expression if necessary.

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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.