# How do you differentiate #y=x^(1/lnx)#?

For this problem, we can use logarithmic differentiation.

Taking the natural logarithm of both sides gives

Now, taking the derivative of both sides yields

Thus

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To differentiate ( y = x^{1/\ln(x)} ), we can use logarithmic differentiation. First, take the natural logarithm of both sides of the equation to simplify the expression:

[ \ln(y) = \ln\left(x^{1/\ln(x)}\right) ]

Using the property ( \ln(a^b) = b \cdot \ln(a) ), we get:

[ \ln(y) = \frac{1}{\ln(x)} \cdot \ln(x) ]

[ \ln(y) = 1 ]

Now, differentiate both sides with respect to ( x ):

[ \frac{1}{y} \cdot \frac{dy}{dx} = 0 ]

Since ( \ln(y) = 1 ), ( y = e^1 = e ). Therefore, ( \frac{1}{y} = \frac{1}{e} ).

So, the derivative of ( y = x^{1/\ln(x)} ) is ( \frac{1}{e} ).

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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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