Question

Simplify the difference quotients f(x+h)-f(x)/h and f(x)-f(a)/x-a for the following function by rationalizing the numerator.

f(x)=10√x​
f(x+h)-f(x)/h =? (Simplify your answer.)

Answer 1

To simplify the difference quotient f(x+h)-f(x)/h for the function f(x)=10√x, substitute the function values into the difference quotient formula. Then rationalize the numerator by multiplying both the numerator and denominator by the conjugate of the numerator.

Step-by-step explanation:

To simplify the difference quotient f(x+h)-f(x)/h for the function f(x)=10√x, we need to substitute the function values into the difference quotient formula. Here's the step-by-step process:

1. Plug in x+h for x in the function: f(x+h) = 10√(x+h)
2. Plug in x for x in the function: f(x) = 10√x
3. Calculate the numerator by subtracting the second equation from the first: f(x+h)-f(x) = 10√(x+h) - 10√x
4. Divide the numerator by h

To rationalize the numerator, we can multiply both the numerator and the denominator by the conjugate of the numerator to eliminate the square root. The conjugate of 10√(x+h) - 10√x is 10√(x+h) + 10√x.

So, the simplified difference quotient with a rationalized numerator is: (10√(x+h) - 10√x) / h * (10√(x+h) + 10√x)