Slope of the tangent line using limit

Question

Use the limit definition of the derivative to find the instantaneous rate of change of  at 

Answer 1

The limit definition is:

f(x) = 2x² + 5x + 3

f(x + h) = 2(x + h)² + 5(x + h) + 3 = 2(x² + 2xh + h²) + 5x + 5h + 3 = 2x² + 4xh + 2h² + 5x + 5h + 3

So f(x + h) - f(x) = 2x² + 4xh + 2h² + 5x + 5h + 3 - (2x² + 5x + 3) = 2x² + 4xh + 2h² + 5x + 5h + 3 - 2x² - 5x - 3 = 4xh + 2h² + 5h = h(4x + 2h + 5)

Then [f(x + h) - f(x)]/h = h(4x + 2h + 5)/h = 4x + 2h + 5

And the limit as h approaches zero of 4x + 2h + 5 = 4x + 5

So f '(x) = 4x + 5