What are the steps in finding the diff quotient for f(x)= sq rt of x?
To find the difference quotient for the function f(x) = √x, which represents the square root of x, you can follow these steps:
Step 1: Start with the given function, f(x) = √x.
Step 2: Determine the formula for the difference quotient. The difference quotient is the average rate of change of f(x) as x changes by a small amount h. It can be calculated using the formula:
[f(x+h) - f(x)] / h.
Step 3: Replace f(x) in the difference quotient formula with the given function f(x) = √x.
[√(x+h) - √x] / h.
Step 4: Simplify the difference quotient expression. The simplification process can involve a variety of algebraic manipulations.
To simplify the given expression [√(x+h) - √x] / h, we can use the conjugate pair technique. Multiplying the numerator and denominator by the conjugate of the numerator, √(x+h) + √x, will help eliminate the square roots.
([√(x+h) - √x] / h) * ([√(x+h) + √x] / √(x+h) + √x).
Step 5: Simplify the expression further by applying the difference of squares formula, (a+b)(a-b) = a^2 - b^2.
[(x+h) - x] / (h * (√(x+h) + √x)),
(x + h - x) / (h * (√(x+h) + √x)),
h / (h * (√(x+h) + √x)).
Step 6: Cancel out common factors. Notice that h appears both in the numerator and denominator, so they cancel each other out.
1 / (√(x+h) + √x).
Therefore, the difference quotient for f(x) = √x is 1 / (√(x+h) + √x).