Differentiate using the product rule

$g(x) = (x + 2 \sqrt x)e^x$

To find the derivative of a function like "g(x) = (x + 2 times the square root of x) times e raised to the power of x", we can use the product rule and chain rule, which are standard techniques in calculus.

The product rule applies because we have two parts being multiplied: "x + 2 times the square root of x" and "e raised to the power of x." The product rule tells us that to find the derivative of two functions multiplied together, we take the derivative of the first part, multiply it by the second part as it is, and then add the first part unchanged multiplied by the derivative of the second part.

For the first part, "x + 2 times the square root of x," we can differentiate each piece separately. The derivative of x is straightforward, and for "2 times the square root of x," we use the chain rule, which helps with differentiating functions involving roots or powers of x. Once we finish differentiating this part, we move on to the second part, "e raised to the power of x," which is simple because its derivative is just the same function. Finally, we combine everything using the product rule to find the complete derivative.

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