Find and simplify the difference quotient f(x + h) - f(x)/h where h is not equal to zero for the following function f(x) = x² - 5.

Solution:

Given, the function is f(x) = x² - 5

We have to find the difference quotient for the function.

f(x + h) = (x + h)² - 5

We know, (a + b)² = a² + b² + 2ab

Now expanding (x + h)² we get,

f(x + h) = x² + 2xh + h² - 5

So, [f(x + h) - f(x)]/h = [(x² + 2xh + h² - 5) - (x² - 5)]/h

= [x² + 2xh + h² - 5 - x² + 5]/h

= [2xh + h²]/h

Taking out h as common in numerator,

[f(x + h) - f(x)]/h = [h(2x + h)]/h

= 2x + h

Therefore, the difference quotient is 2x + h.

Find and simplify the difference quotient f(x + h) - f(x)/h where h is not equal to zero for the following function f(x) = x² - 5.

Summary:

The difference quotient f(x + h) - f(x)/h, where h is not equal to zero for the following function f(x) = x² - 5 is 2x + h.