# How do you differentiate #g(x) = x^3e^(2x)# using the product rule?

As per the product rule,

Thus,

Determine every derivative.

The chain rule will be needed for this:

Factored optionally:

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To differentiate g(x) = x^3e^(2x) using the product rule:

- Identify the functions u(x) and v(x). Let u(x) = x^3 and v(x) = e^(2x).
- Apply the product rule, which states that the derivative of the product of two functions is equal to the derivative of the first function times the second function, plus the first function times the derivative of the second function.
- Calculate the derivatives of u(x) and v(x).
- Use the product rule formula to find the derivative of g(x), which is g'(x).
- Substitute the derivatives of u(x) and v(x) into the product rule formula.
- Simplify the expression to get the final result for g'(x).

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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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