# What is the derivative of #y=5^(-1/x)#?

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To find the derivative of ( y = 5^{-\frac{1}{x}} ), we can use the chain rule.

Let ( u = -\frac{1}{x} ).

( \frac{du}{dx} = \frac{d}{dx}(-\frac{1}{x}) )

( \frac{du}{dx} = \frac{d}{dx}(-x^{-1}) )

( \frac{du}{dx} = -(-1)x^{-1-1} )

( \frac{du}{dx} = x^{-2} )

Now, applying the chain rule:

( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} )

( \frac{dy}{dx} = \frac{d}{du}(5^u) \cdot x^{-2} )

( \frac{dy}{dx} = 5^u \cdot \ln(5) \cdot x^{-2} )

Substituting back ( u = -\frac{1}{x} ):

( \frac{dy}{dx} = 5^{-\frac{1}{x}} \cdot \ln(5) \cdot x^{-2} )

So, the derivative of ( y = 5^{-\frac{1}{x}} ) is ( \frac{-\ln(5)}{x^2 \cdot 5^{1/x}} ).

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