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

The rule for differentiating e functions is,

so in the case of your question we have

So now what is y, f(x) and g(x).

so subbing into the original formula,

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To find the derivative of ( y = e^{e^{3x^2}} ), we use the chain rule. The chain rule states that if we have a function within another function, we take the derivative of the outer function with the inner function intact, then multiply by the derivative of the inner function.

Let ( u = e^{3x^2} ). Then, ( y = e^u ).

Taking the derivative of ( y ) with respect to ( x ) using the chain rule:

[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} ]

Where:

[ \frac{dy}{du} = e^u ] [ \frac{du}{dx} = \frac{d}{dx}(e^{3x^2}) = 6x \cdot e^{3x^2} ]

Therefore:

[ \frac{dy}{dx} = e^{e^{3x^2}} \cdot (6x \cdot e^{3x^2}) = 6x \cdot e^{3x^2} \cdot e^{e^{3x^2}} ]

So, the derivative of ( y ) with respect to ( x ) is ( 6x \cdot e^{3x^2} \cdot e^{e^{3x^2}} ).

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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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