# How do you find the first and second derivatives of #f(x)=cos(x)/x^8 # using the quotient rule?

First derivative:

Second derivative:

So, going back to the exercise we have:

Thus:

Factoring:

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To find the first derivative of ( f(x) = \frac{\cos(x)}{x^8} ) using the quotient rule:

- Apply the quotient rule: ( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{vu' - uv'}{v^2} )
- Let ( u = \cos(x) ) and ( v = x^8 ).
- Find the derivatives ( u' ) and ( v' ):
- ( u' = -\sin(x) )
- ( v' = 8x^7 )

- Substitute into the quotient rule formula: ( f'(x) = \frac{x^8(-\sin(x)) - \cos(x)(8x^7)}{(x^8)^2} ) ( f'(x) = \frac{-x^8\sin(x) - 8x^7\cos(x)}{x^{16}} ) ( f'(x) = \frac{-\sin(x)x^8 - 8x^7\cos(x)}{x^{16}} )

To find the second derivative, differentiate ( f'(x) ) using the quotient rule again:

- Apply the quotient rule: ( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{vu' - uv'}{v^2} )
- Let ( u = -\sin(x)x^8 - 8x^7\cos(x) ) and ( v = x^{16} ).
- Find the derivatives ( u' ) and ( v' ):
- ( u' = -\cos(x)x^8 - 8x^7\sin(x) - 56x^6\cos(x) )
- ( v' = 16x^{15} )

- Substitute into the quotient rule formula: ( f''(x) = \frac{x^{16}(-\cos(x)x^8 - 8x^7\sin(x) - 56x^6\cos(x)) - (-\sin(x)x^8 - 8x^7\cos(x))(16x^{15})}{(x^{16})^2} ) ( f''(x) = \frac{-x^{24}\cos(x) - 8x^{23}\sin(x) - 56x^{22}\cos(x) + 128x^{22}\cos(x) + 128x^{22}\sin(x)}{x^{32}} ) ( f''(x) = \frac{-x^{24}\cos(x) - 8x^{23}\sin(x) - 56x^{22}\cos(x) + 128x^{22}\cos(x) + 128x^{22}\sin(x)}{x^{32}} )

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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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