Simplify the difference quotients ƒ(x+h) - ƒ(x) / h and ƒ(x) - ƒ(a) / (x-a) by rationalizing the numerator.
ƒ(x) = - (3/√x)

Verified step by step guidance

1. Step 1: Identify the function f(x) = -\frac{3}{\sqrt{x}} and the expressions to simplify: \frac{f(x+h) - f(x)}{h} and \frac{f(x) - f(a)}{x-a}.
2. Step 2: For the first expression \frac{f(x+h) - f(x)}{h}, substitute f(x+h) = -\frac{3}{\sqrt{x+h}} and f(x) = -\frac{3}{\sqrt{x}} into the expression.
3. Step 3: Simplify the numerator of \frac{-\frac{3}{\sqrt{x+h}} + \frac{3}{\sqrt{x}}}{h} by finding a common denominator, which is \sqrt{x+h}\sqrt{x}.
4. Step 4: Rationalize the numerator by multiplying both the numerator and the denominator by the conjugate of the numerator, which is \sqrt{x+h}\sqrt{x}.
5. Step 5: For the second expression \frac{f(x) - f(a)}{x-a}, substitute f(x) = -\frac{3}{\sqrt{x}} and f(a) = -\frac{3}{\sqrt{a}}, and rationalize the numerator by multiplying by the conjugate \sqrt{x}\sqrt{a}.