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 of the derivative is:

For 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)

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

f(x + h) - f(x) = 4xh + 2h² + 5h

f(x + h) - f(x) = h(4x + 2h + 5)

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

And the limit as h approaches zero is 4x + 5

So the instantaneous rate of change of f(x) at x = 4 is f '(4) = 4(4) + 5 = 21