# How do you find the 1st and 2nd derivative of #2e^(2x)#?

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To find the derivative of (2e^{2x}), we can use the chain rule, which states that if (f(x)) and (g(x)) are differentiable functions, then the derivative of (f(g(x))) with respect to (x) is (f'(g(x)) \cdot g'(x)).

The first derivative of (2e^{2x}) is: [ \frac{d}{dx} (2e^{2x}) = 2 \cdot \frac{d}{dx}(e^{2x}) ]

Using the chain rule, the derivative of (e^{2x}) with respect to (x) is: [ \frac{d}{dx}(e^{2x}) = e^{2x} \cdot \frac{d}{dx}(2x) ]

[ = e^{2x} \cdot 2 ]

So, the first derivative becomes: [ 2 \cdot 2e^{2x} = 4e^{2x} ]

Now, to find the second derivative, we take the derivative of (4e^{2x}) with respect to (x): [ \frac{d}{dx} (4e^{2x}) = 4 \cdot \frac{d}{dx}(e^{2x}) ]

[ = 4 \cdot 2e^{2x} ]

[ = 8e^{2x} ]

Therefore, the first derivative of (2e^{2x}) is (4e^{2x}), and the second derivative is (8e^{2x}).

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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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