# How do you differentiate #e^(lnx) #?

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To differentiate ( e^{\ln(x)} ), you can use the chain rule. The derivative is:

[ \frac{d}{dx} \left( e^{\ln(x)} \right) = e^{\ln(x)} \cdot \frac{d}{dx}(\ln(x)) ]

Using the chain rule and the derivative of (\ln(x)) with respect to (x), which is (1/x), the expression simplifies to:

[ \frac{d}{dx} \left( e^{\ln(x)} \right) = e^{\ln(x)} \cdot \frac{1}{x} ]

Thus, the derivative of (e^{\ln(x)}) is ( \frac{1}{x} \cdot e^{\ln(x)} ), or simply ( \frac{e^{\ln(x)}}{x} ).

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To differentiate ( e^{\ln(x)} ), we can use the chain rule. The chain rule states that if we have a function ( f(g(x)) ), then its derivative is given by ( f'(g(x)) \times g'(x) ).

In this case, ( f(g(x)) = e^{\ln(x)} ) and ( g(x) = \ln(x) ).

The derivative of ( e^{\ln(x)} ) can be found as follows:

[ \frac{d}{dx} \left( e^{\ln(x)} \right) = e^{\ln(x)} \times \frac{d}{dx}(\ln(x)) ]

Now, we need to find the derivative of ( \ln(x) ). The derivative of ( \ln(x) ) is ( \frac{1}{x} ).

Substituting this into our expression, we get:

[ \frac{d}{dx} \left( e^{\ln(x)} \right) = e^{\ln(x)} \times \frac{1}{x} ]

[ = \frac{e^{\ln(x)}}{x} ]

So, the derivative of ( e^{\ln(x)} ) with respect to ( x ) is ( \frac{e^{\ln(x)}}{x} ).

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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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