How do you find the first and second derivatives of #y = (x^2 + 3) / e^x# using the quotient rule?
Alternatively, you could rewrite the original question as a product and apply the product rule instead of the quotient rule.
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To find the first and second derivatives of ( y = \frac{x^2 + 3}{e^{x}} ) using the quotient rule:

First, let's denote ( u = x^2 + 3 ) and ( v = e^{x} ).

Then, we'll find the derivatives of ( u ) and ( v ):
( u' = 2x )
( v' = e^{x} )

Next, we'll apply the quotient rule, which states that for functions ( u(x) ) and ( v(x) ), the derivative of their quotient is:
( \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v  uv'}{v^2} )

Now, using the quotient rule, we have:
( \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{(2x)(e^{x})  (x^2 + 3)(e^{x})}{(e^{x})^2} )

Simplifying this expression gives us the first derivative:
( \frac{d}{dx}\left(\frac{x^2 + 3}{e^{x}}\right) = \frac{2xe^{x} + (x^2 + 3)e^{x}}{e^{2x}} )

To find the second derivative, we need to differentiate the first derivative again with respect to ( x ):
( \frac{d^2}{dx^2}\left(\frac{x^2 + 3}{e^{x}}\right) = \frac{d}{dx}\left(\frac{2xe^{x} + (x^2 + 3)e^{x}}{e^{2x}}\right) )

Simplify the expression obtained from the first derivative and then apply the quotient rule again to find the second derivative.

After simplifying the expression obtained for the second derivative, you'll have the result.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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