# How do you differentiate #(1)/ (x^2cosx)# using the quotient rule?

The answer is

The rule of the quotient is

additionally, the rule for products is

This is what we have

Consequently,

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To differentiate ( \frac{1}{x^2 \cos(x)} ) using the quotient rule, you can follow these steps:

- Let ( u = 1 ) and ( v = x^2 \cos(x) ).
- Compute ( u' ) (the derivative of ( u )) as 0, since the derivative of a constant is zero.
- Compute ( v' ) (the derivative of ( v )) using the product rule: ( v' = (2x)(\cos(x)) + (x^2)(-\sin(x)) ).
- Apply the quotient rule: ( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{(u'v - uv')}{v^2} ).

Plugging in the values:

- ( u' = 0 )
- ( v' = 2x \cos(x) - x^2 \sin(x) )
- ( u = 1 )
- ( v = x^2 \cos(x) )

Using the quotient rule formula:

( \frac{d}{dx} \left( \frac{1}{x^2 \cos(x)} \right) = \frac{(0)(x^2 \cos(x)) - (1)((2x \cos(x)) - (x^2 \sin(x)))}{(x^2 \cos(x))^2} )

Simplifying further:

( \frac{d}{dx} \left( \frac{1}{x^2 \cos(x)} \right) = \frac{-2x \cos(x) + x^2 \sin(x)}{(x^2 \cos(x))^2} )

This is the derivative of ( \frac{1}{x^2 \cos(x)} ) using the quotient rule.

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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.

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