To verify if a function $f$ is derivable (differentiable) on an interval, you generally follow a series of steps. Here's how you can approach the process:

Steps to Verify if $f$ is Derivable on an Interval:

1. Check Continuity of $f$ on the Interval:

   • A function must be continuous on an interval before it can be differentiable there. So, verify that the function $f$ is continuous on the given interval $(a, b)$ or $[a, b]$.
   • This means:
     • $\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = f(c)$ for every point $c$ in the interval.

2. Existence of the Derivative at Each Point:

   • The function is differentiable at a point $c$ if the derivative exists at that point.
   • Mathematically, you compute the limit: $f'(c) = \lim_{h \to 0} \frac{f(c + h) - f(c)}{h}$ If this limit exists for all points in the interval, then the function is differentiable on that interval.

3. Check for Differentiability at the Boundary Points (if applicable):

   • If the interval is closed, i.e., $[a, b]$, then you need to check the differentiability at the boundary points $a$ and $b$.
   • For differentiability at the endpoint $a$, check: $f'(a) = \lim_{h \to 0^+} \frac{f(a + h) - f(a)}{h}$ Similarly for $b$, check: $f'(b) = \lim_{h \to 0^-} \frac{f(b + h) - f(b)}{h}$

4. Examine Possible Discontinuities or Non-Smooth Behavior:

   • If there are any points where the function has corners (sharp turns), vertical tangents, or discontinuities, the function will not be differentiable at those points. These features must be examined to confirm differentiability.

5. Use Standard Derivability Rules:

   • If $f$ is composed of standard differentiable functions (such as polynomials, trigonometric functions, exponentials, etc.), you can use known rules to conclude differentiability. For example, sums, products, and quotients of differentiable functions are also differentiable (with exceptions for division by zero).

Example:

• Consider the function $f(x) = x^2$ on the interval $[0, 2]$:
  • First, check if $f(x)$ is continuous on the interval. Since $x^2$ is a polynomial, it is continuous everywhere.
  • Then, compute the derivative: $f'(x) = 2x$
  • The derivative $f'(x)$ exists everywhere on the interval, so $f(x)$ is differentiable on $[0, 2]$.

Key Criteria for Derivability:

1. Continuity of the function on the interval.
2. The existence of the limit defining the derivative at every point.
3. The absence of sharp turns, cusps, or discontinuities within the interval.
4. Differentiability at the endpoints (if the interval is closed).

Would you like to see examples or have further details about any of these steps?

Here are 5 related questions to explore:

1. How do you prove that a function is continuous on an interval?
2. What is the relationship between differentiability and continuity?
3. How to handle piecewise functions for differentiability?
4. Can a function be differentiable but not continuous?
5. How to find the derivative using first principles?

Tip: A useful way to check differentiability at endpoints is to carefully compute the one-sided limits for the derivative at those points.