How do you differentiate #y=(1+x)^(1/x)#?
# dy/dx = (1+x)^(1/x) {1/(x(1+x))  ln(1+x)/x^2} #
We have:
Differentiating implicitly and applying the quotient rule and the chain rule gives:
And so:
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The answer is
We diiferentiate using logs.
Differentiating
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To differentiate ( y = (1 + x)^{\frac{1}{x}} ), we will use the logarithmic differentiation method.

Take the natural logarithm of both sides: [ \ln(y) = \ln\left((1 + x)^{\frac{1}{x}}\right) ] [ \ln(y) = \frac{1}{x} \ln(1 + x) ]

Differentiate implicitly with respect to ( x ): [ \frac{1}{y} \cdot \frac{dy}{dx} = \frac{d}{dx}\left(\frac{1}{x} \ln(1 + x)\right) ]

Use the product rule and the chain rule to differentiate the righthand side: [ \frac{1}{y} \cdot \frac{dy}{dx} = \frac{d}{dx}\left(\frac{1}{x}\right) \cdot \ln(1 + x) + \frac{1}{x} \cdot \frac{d}{dx}(\ln(1 + x)) ] [ \frac{1}{y} \cdot \frac{dy}{dx} = \frac{1}{x^2} \cdot \ln(1 + x) + \frac{1}{x} \cdot \frac{1}{1 + x} ]

Simplify the expression: [ \frac{1}{y} \cdot \frac{dy}{dx} = \frac{\ln(1 + x)}{x^2} + \frac{1}{x(1 + x)} ]

Now, find ( \frac{dy}{dx} ) by multiplying both sides by ( y ) (which is ( (1 + x)^{\frac{1}{x}} )): [ \frac{dy}{dx} = y \left(\frac{\ln(1 + x)}{x^2} + \frac{1}{x(1 + x)}\right) ] [ \frac{dy}{dx} = (1 + x)^{\frac{1}{x}} \left(\frac{\ln(1 + x)}{x^2} + \frac{1}{x(1 + x)}\right) ]
Therefore, the derivative of ( y = (1 + x)^{\frac{1}{x}} ) is ( (1 + x)^{\frac{1}{x}} \left(\frac{\ln(1 + x)}{x^2} + \frac{1}{x(1 + x)}\right) ).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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