Keep in mind that there are infinitely many tangents for any
curve because there are infinitely many points on the curve.How do we perform this shrinking act? By using the limits we discussed in Chapter 3. We set up a limit
during which h approaches zero, like the following:
This is the definition of the derivative, and we call it f′(x).
Notice that the equation is just a slightly modified version of the difference quotient, with different
notation. The only difference is that we’re finding the slope between two points that are infinitesimally
close to each other.
Example 1: Find the slope of the curve f(x) = x^2 at the point (2, 4).
This means that x 1 = 2 and f(2) = 2^2 = 4. If we can figure out f(x 1 + h), then we can find the slope. Well,
how did we find the value of f(x)? We plugged x 1 into the equation f(x) = x^2 . To find f(x 1 + h) we plug x 1
- h into the equation, which now looks like this
f(x 1 + h) = (2 + h)^2 = 4 + 4h + h^2Now plug this into the slope formula.
Next simplify by factoring h out of the top.
Taking the limit as h approaches 0, we get 4. Therefore, the slope of the curve y = x^2 at the point (2, 4) is
- Now we’ve found the slope of a curve at a certain point, and the notation looks like this: f ́(2) = 4.
Remember this notation!
Example 2: Find the derivative of the equation in Example 1 at the point (5, 25). This means that x 1 = 5