and f(x) = 25. This time,
(x 1 + h)^2 = (5 + h)^2 = 25 + 10h + h^2Now plug this into the formula for the derivative.
Once again, simplify by factoring h out of the top.
Taking the limit as h goes to 0, you get 10. Therefore, the slope of the curve y = x^2 at the point (5, 25) is
10, or: f ́(5) = 10.
Using this pattern, let’s forget about the arithmetic for a second and derive a formula.
Example 3: Find the slope of the equation f(x) = x^2 at the point (x 1 , x 12 ).
Follow the steps in the last two problems, but instead of using a number, use x 1 . This means that f(x 1 ) =
x 12 and (x 1 + h)^2 = x 12 + 2x 1 h + h^2 . Then the derivative is
Factor h out of the top.
Now take the limit as h goes to 0: you get 2x 1 . Therefore, f′(x 1 ) = 2x 1.
This example gives us a general formula for the derivative of this curve. Now we can pick any point, plug
it into the formula, and determine the slope at that point. For example, the derivative at the point x = 7 is
- At the point x = , the derivative is .