# How do you differentiate #y=lnx/x^20#?

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To differentiate ( y = \frac{\ln x}{x^{20}} ), you can use the quotient rule:

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

Let ( f(x) = \ln x ) and ( g(x) = x^{20} ).

[ f'(x) = \frac{1}{x} ] [ g'(x) = 20x^{19} ]

Apply the quotient rule:

[ \frac{d}{dx} \left( \frac{\ln x}{x^{20}} \right) = \frac{\frac{1}{x} \cdot x^{20} - \ln x \cdot 20x^{19}}{(x^{20})^2} ]

Simplify:

[ \frac{d}{dx} \left( \frac{\ln x}{x^{20}} \right) = \frac{x^{20} - 20x^{19}\ln x}{x^{40}} ]

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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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