# How do you find the derivative of #y=(e^(5x^4))/(e^(4x^2+3))#?

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To find the derivative of ( y = \frac{e^{5x^4}}{e^{4x^2+3}} ), you can use the quotient rule.

The quotient rule states that if you have a function ( y = \frac{u}{v} ), then its derivative is given by ( y' = \frac{u'v - uv'}{v^2} ).

First, find the derivatives of ( u ) and ( v ) separately. Then apply the quotient rule to find the derivative of ( y ).

Let ( u = e^{5x^4} ) and ( v = e^{4x^2+3} ).

( u' = \frac{d}{dx}(e^{5x^4}) ) and ( v' = \frac{d}{dx}(e^{4x^2+3}) ).

Using the chain rule, ( u' = 20x^3 e^{5x^4} ) and ( v' = (8x)e^{4x^2+3} ).

Now, apply the quotient rule:

[ y' = \frac{(20x^3 e^{5x^4})(e^{4x^2+3}) - (e^{5x^4})(8x)e^{4x^2+3}}{(e^{4x^2+3})^2} ]

Simplify this expression to get the derivative of ( 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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