# What is the derivative of #x^(3/x)#?

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To find the derivative of (x^{3/x}), you can use the logarithmic differentiation method.

Let (y = x^{3/x}).

Take the natural logarithm of both sides:

[\ln(y) = \ln(x^{3/x})]

Apply the power rule of logarithms:

[\ln(y) = \frac{3}{x} \ln(x)]

Differentiate implicitly with respect to (x):

[\frac{1}{y} \frac{dy}{dx} = \frac{d}{dx} \left(\frac{3}{x} \ln(x)\right)]

Using the product rule and the chain rule:

[\frac{1}{y} \frac{dy}{dx} = \frac{3}{x^2} \ln(x) - \frac{3}{x^2}]

Now, substitute back (y = x^{3/x}):

[\frac{1}{x^{3/x}} \frac{dy}{dx} = \frac{3}{x^2} \ln(x) - \frac{3}{x^2}]

Multiply both sides by (x^{3/x}):

[\frac{dy}{dx} = x^{3/x} \left(\frac{3}{x^2} \ln(x) - \frac{3}{x^2}\right)]

Simplify:

[\frac{dy}{dx} = 3x^{1 - \frac{3}{x}} (\ln(x) - 1)]

So, the derivative of (x^{3/x}) is (3x^{1 - \frac{3}{x}} (\ln(x) - 1)).

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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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