How do you differentiate #f(x)= (2 x^2 + 7 x  2)/ (x  cos x )# using the quotient rule?
We are required to locate the derivative.
Apply the rule of quotients, which is
where
which is also able to be written as
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To differentiate the function ( f(x) = \frac{2x^2 + 7x  2}{x  \cos x} ) using the quotient rule, follow these steps:

Apply the quotient rule, which states that the derivative of ( \frac{u}{v} ) is ( \frac{u'v  uv'}{v^2} ).

Identify ( u ) and ( v ): ( u = 2x^2 + 7x  2 ) ( v = x  \cos x )

Find the derivatives of ( u ) and ( v ): ( u' = 4x + 7 ) ( v' = 1 + \sin x )

Apply the quotient rule formula: ( f'(x) = \frac{(u'v  uv')}{v^2} )

Substitute the values of ( u ), ( v ), ( u' ), and ( v' ) into the formula: ( f'(x) = \frac{((4x + 7)(x  \cos x)  (2x^2 + 7x  2)(1 + \sin x))}{(x  \cos x)^2} )

Simplify the expression as needed.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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