# How do you find the derivative for #y= cos(x)/x^8#?

You can use the *quotient rule* or the *product rule*.

You actually have two ways of approaching this derivative.

You can thus differentiate this function by using

The quotient rule allows you to calculate the derivative of a function expressed as a quotient of two other functions by using the

This will get you

This is equivalent to

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To find the derivative of ( y = \frac{\cos(x)}{x^8} ), you can use the quotient rule. The quotient rule states that if you have a function in the form ( \frac{f(x)}{g(x)} ), then its derivative is given by:

[ \frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2} ]

Using this rule, where ( f(x) = \cos(x) ) and ( g(x) = x^8 ), and knowing that the derivative of ( \cos(x) ) is ( -\sin(x) ), and the derivative of ( x^8 ) is ( 8x^7 ), you can compute the derivative of ( y ).

[ y' = \frac{d}{dx} \left( \frac{\cos(x)}{x^8} \right) = \frac{-\sin(x)(x^8) - (\cos(x))(8x^7)}{(x^8)^2} ]

Simplify the expression to obtain the derivative of ( y ).

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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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