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

Here, we have used the Chain Rule & the Quotient Rule.

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To find the derivative of ( y = (1+x)^{\frac{1}{x}} ), you can use logarithmic differentiation. The steps are as follows:

- Take the natural logarithm of both sides: ( \ln(y) = \ln((1+x)^{\frac{1}{x}}) ).
- Apply the logarithmic properties to simplify: ( \ln(y) = \frac{1}{x} \ln(1+x) ).
- Differentiate both sides implicitly with respect to ( x ).
- Solve for ( y' ), the derivative of ( y ) with respect to ( x ).

The derivative ( y' ) will be expressed in terms of ( x ) and ( y ), the original function.

Would you like me to provide the derivative explicitly, or do you need further explanation on any step?

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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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