# How do you find the derivative of #e^(x(3x^2 + 2x-1)^2#?

The answer is:

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To find the derivative of ( e^{x(3x^2 + 2x - 1)^2} ), you would use the chain rule. The derivative can be calculated as follows:

Let ( u = x(3x^2 + 2x - 1)^2 ). Then, ( y = e^u ).

Apply the chain rule:

( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} )

Now, differentiate ( y = e^u ) with respect to ( u ) to get ( \frac{dy}{du} = e^u ).

Differentiate ( u = x(3x^2 + 2x - 1)^2 ) with respect to ( x ) to get ( \frac{du}{dx} = (3x^2 + 2x - 1)^2 + 2x(3x^2 + 2x - 1)(6x + 2) ).

Now, substitute these derivatives back into the chain rule formula:

( \frac{dy}{dx} = e^u \cdot ((3x^2 + 2x - 1)^2 + 2x(3x^2 + 2x - 1)(6x + 2)) )

Finally, substitute back the expression for ( u ):

( \frac{dy}{dx} = e^{x(3x^2 + 2x - 1)^2} \cdot ((3x^2 + 2x - 1)^2 + 2x(3x^2 + 2x - 1)(6x + 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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