# How do you find the derivative of #y=1+x^-1+x^-2+x^-3#?

The first derivative is

Use the power rule:

Here's the derivative worked out (I color-coded some parts so that they would be easier to follow):

That's the derivative. Hope this helped!

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To find the derivative of ( y = 1 + x^{-1} + x^{-2} + x^{-3} ), you differentiate each term separately using the power rule, which states that ( \frac{d}{dx}(x^n) = nx^{n-1} ), where ( n ) is the exponent:

[ \frac{dy}{dx} = 0 - x^{-2} -2x^{-3} -3x^{-4} ]

Simplify the expression:

[ \frac{dy}{dx} = -x^{-2} -2x^{-3} -3x^{-4} ]

[ \frac{dy}{dx} = -\frac{1}{x^2} -\frac{2}{x^3} -\frac{3}{x^4} ]

So, the derivative of ( y = 1 + x^{-1} + x^{-2} + x^{-3} ) is ( \frac{dy}{dx} = -\frac{1}{x^2} -\frac{2}{x^3} -\frac{3}{x^4} ).

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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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