# How do you differentiate #g(x) = (1/x^3)*sqrt(x-e^(2x))# using the product rule?

So, we have

then the derivative of

So, it's equal to

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To differentiate ( g(x) = \frac{1}{x^3} \sqrt{x - e^{2x}} ) using the product rule, follow these steps:

- Identify the two functions being multiplied: ( f(x) = \frac{1}{x^3} ) and ( h(x) = \sqrt{x - e^{2x}} ).
- Apply the product rule: ( g'(x) = f'(x)h(x) + f(x)h'(x) ).
- Calculate the derivatives of ( f(x) ) and ( h(x) ).
- ( f'(x) = -\frac{3}{x^4} )
- ( h'(x) = \frac{1}{2\sqrt{x - e^{2x}}} \cdot (1 - 2e^{2x}) )

- Substitute the derivatives and the original functions into the product rule formula.

[ g'(x) = -\frac{3}{x^4} \cdot \sqrt{x - e^{2x}} + \frac{1}{x^3} \cdot \frac{1}{2\sqrt{x - e^{2x}}} \cdot (x - e^{2x}) ]

- Simplify the expression if necessary.

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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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