Find the Difference Quotient f(x) = square root of x

Problem

ƒ(x)=√(,x)

Solution

1. State the formula for the difference quotient of a function ƒ(x)

(ƒ*(x+h)−ƒ(x))/h

2. Substitute the function ƒ(x)=√(,x) into the formula.

(√(,x+h)−√(,x))/h

3. Rationalize the numerator by multiplying the numerator and the denominator by the conjugate √(,x+h)+√(,x)

(√(,x+h)−√(,x))/h⋅(√(,x+h)+√(,x))/(√(,x+h)+√(,x))

4. Expand the numerator using the difference of squares (a−b)*(a+b)=a^2−b^2

((x+h)−x)/(h*(√(,x+h)+√(,x)))

5. Simplify the numerator by subtracting x from x

h/(h*(√(,x+h)+√(,x)))

6. Divide out the common factor of h from the numerator and denominator.

1/(√(,x+h)+√(,x))

Final Answer

(ƒ*(x+h)−ƒ(x))/h=1/(√(,x+h)+√(,x))

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