Derivative of √x √x
Understanding the Product Rule:
The derivative of a product of two functions can be found using 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.

Applying the Product Rule:
In the case of √x √x, we can treat it as the product of two functions: f(x) = √x and g(x) = √x.
Now, according to the product rule, the derivative of √x √x can be calculated as follows:
f'(x) = (d/dx)(√x) * √x + √x * (d/dx)(√x)

Finding the Derivative:
To find the derivative of √x, we can rewrite it as x^(1/2) and then apply the power rule.
(d/dx)(√x) = (1/2)x^(-1/2)
Now we can substitute this derivative back into our equation:
f'(x) = (1/2)x^(-1/2) * √x + √x * (1/2)x^(-1/2)
Simplify the expression by combining like terms:
f'(x) = (1/2√x) + (1/2√x)
f'(x) = √x

Conclusion:
Therefore, the derivative of √x √x is simply √x. This result can be obtained by applying the product rule and simplifying the expression.