How do you differentiate #f(x)= (2 x^2 + 7 x - 2)/ (x sinx )# using the quotient rule?

Answer 1

The Quotient Rule says #d/dx(f(x)/g(x))=(g(x) * f'(x) - f(x) * g'(x))/((g(x))^2)#. This, in addition to the product rule, linearity, and the power rule, gives:

#d/dx((2x^2+7x-2)/(x sin(x)))=(x sin(x) * (4x+7) - (2x^2+7x-2) * (sin(x) + x cos(x)))/(x^2 sin^{2}(x))#

#=(2x^{2}sin(x)+2sin(x)-2x^{3}cos(x)-7x^{2}cos(x)+2xcos(x))/(x^2 sin^{2}(x))#

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Answer 2

Evelyn Agard

To differentiate the function ( f(x) = \frac{2x^2 + 7x - 2}{x \sin(x)} ) using the quotient rule, follow these steps:

Identify ( u(x) = 2x^2 + 7x - 2 ) and ( v(x) = x \sin(x) ).
Apply the quotient rule formula: ( \frac{d}{dx}\left(\frac{u(x)}{v(x)}\right) = \frac{v(x)u'(x) - u(x)v'(x)}{(v(x))^2} ).
Compute ( u'(x) ) and ( v'(x) ) using the power rule and product rule respectively.
Substitute the values into the quotient rule formula and simplify the expression.

The derivative of ( f(x) ) will be the result of this computation.

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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.