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

We use the derivative of the exponential, the chain rule and the quotient rule.

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To differentiate ( e^{x/y} ) with respect to ( x ), we use the chain rule. The derivative is calculated as follows:

[ \frac{d}{dx} \left( e^{x/y} \right) = \frac{d}{dx} \left( e^{u} \right) = e^{u} \cdot \frac{du}{dx} ]

where ( u = x/y ). Differentiating ( u ) with respect to ( x ) gives:

[ \frac{du}{dx} = \frac{1}{y} ]

Substituting this back into the expression:

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

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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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