# How do you differentiate #y=e^x/x^7#?

The easiest way, for me, is to first write this not as a quotient:

Thus:

Simplifying:

Common denominator:

Thus:

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To differentiate ( y = \frac{e^x}{x^7} ), you can use the quotient rule. The quotient rule states that for functions ( u ) and ( v ), if ( y = \frac{u}{v} ), then the derivative ( \frac{dy}{dx} ) is given by:

[ \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} ]

In this case, ( u = e^x ) and ( v = x^7 ). So, we need to find the derivatives of ( u ) and ( v ), which are ( \frac{du}{dx} = e^x ) and ( \frac{dv}{dx} = 7x^6 ), respectively.

Applying the quotient rule, we get:

[ \frac{dy}{dx} = \frac{x^7(e^x) - e^x(7x^6)}{(x^7)^2} ]

Simplify this expression to get the final result.

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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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