# How do you differentiate #((e^x)+x)^(1/x)#?

# d/dx (e^x+x)^(1/x)= (e^x+x)^(1/x){ (e^x+1)/(x(e^x+x)) -(ln(e^x+x))/x^2 } #

The best approach here is to use logarithmic differentiation to remove the variable exponent, as follows:

We can now differentiate the LHS implicitly, and the RHS using the product rule to get:

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To differentiate ((e^x)+x)^(1/x), you can use the chain rule. First, take the natural logarithm of the function to simplify it. Then apply the chain rule and differentiate step by step. The result will be a bit complex, involving the original function and its derivatives. Here's the simplified process:

Let y = ((e^x) + x)^(1/x) Take the natural logarithm of both sides: ln(y) = (1/x) * ln((e^x) + x) Differentiate implicitly with respect to x: (dy/dx) / y = [(1/x) * (d/dx) ln((e^x) + x)] - ln((e^x) + x) / x^2

Then solve for dy/dx. The result will involve the original function and its derivatives.

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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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