Find the derivative of the function.
f(x) = sqrt{x} -[3]sqrt{x}
To find the derivative of the function f(x) = sqrt(x) - 3*sqrt(x), we can use the power rule and the chain rule. Here's how:
Step 1: Rewrite the function using exponent notation.
f(x) = x^(1/2) - 3*x^(1/2)
Step 2: Apply the power rule.
The power rule states that if you have a function of the form f(x) = x^n, then the derivative is given by f'(x) = n*x^(n-1).
Applying the power rule to each term, we get:
f'(x) = (1/2)*x^(-1/2) - 3*(1/2)*x^(-1/2)
Step 3: Simplify.
To simplify the expression, we can combine like terms. Notice that both terms have x^(-1/2) as a factor, so we can factor it out:
f'(x) = (1/2)*x^(-1/2) - 3*(1/2)*x^(-1/2)
= (1/2 - 3/2)*x^(-1/2)
= (-2/2)*x^(-1/2)
= -x^(-1/2)
Step 4: Rewrite using radical notation.
Finally, we can rewrite the derivative using radical notation:
f'(x) = -1/sqrt(x)
So, the derivative of f(x) = sqrt(x) - 3*sqrt(x) is f'(x) = -1/sqrt(x).