DIFFERENTIABILITY
One of the important requirements for the differentiability of a function is that the function be continuous.
But, even if a function is continuous at a point, the function is not necessarily differentiable there. Check
out the graph below.
If a function has a “sharp corner,” you can draw more than one tangent line at that point, and because the
slopes of these tangent lines are not equal, the function is not differentiable there.
Another possible problem occurs when the tangent line is vertical (which can also occur at a cusp)
because a vertical line has an infinite slope. For example, if the derivative of a function is , it
doesn’t have a derivative at x = −1.
Try these problems on your own, then check your work against the answers immediately beneath each
problem.
PROBLEM 1. Find the derivative of f(x) = 3x^2 at (4, 48).
Answer: f(4 + h) = 3(4 + h)^2 = 48 + 24h + 3h^2 . Use the definition of the derivative.
Simplify.