To **determine** if a **function** is **differentiable**, I first verify its **continuity** across its entire **domain.**

A **function** f(x) is considered **differentiable** at a point if it has a defined **derivative** at that point, meaning the slope of the **tangent** to the **curve** at that **point exists.**

I check this by calculating the derivative f'(x), and if it exists for all x within the **domain** of the **function**, then the **function** is **differentiable** on that domain.

In **calculus**, **differentiability** is crucial because it allows for the application of **derivative-based** techniques and** fundamental theorems of calculus** which are essential for analyzing the behavior of **functions.**

Understanding the relationship between continuity and **differentiability** is a key step. Although every **differentiable** **function** is continuous, the reverse isn’t always true. For a **function** to be **differentiable** at a point, it not only needs to be continuous there but also smooth, without any sharp corners or cusps.

As I explore further, I’ll remember that **differentiability** ensures a smooth, predictable change – a concept at the heart of **calculus**, enabling a deeper comprehension of **change** and **motion.**

Stay tuned; I’m about to uncover the world where the elegant dance of **curves** and slopes comes to life!

## Analyzing The Function’s Graph For Knowing Differentiability

When I look at a function’s **graph**, I’m seeking to determine if it is **differentiable**. A **differentiable** function typically means that there is a **tangent line** at every point along the **graph**.

For a function to be **differentiable** at a point, the **tangent** must exist, and the function must be **continuous** at that point. Moreover, the **derivative** of the function at that point must exist.

I first check for any **discontinuities**. If the **graph** shows any breaks, holes, or jumps, the function is not **continuous** there, implying non-**differentiability**. Next, the **graph** should not have any **sharp corners** or cusps—points where the **tangent line** is undefined because the **slope** changes abruptly.

Another visual clue for non-**differentiability** includes a **vertical tangent line**, indicating an infinite slope that defies the existence of a real-valued **derivative**. The **limit** of the **slope** of the secant lines should approach the **slope** of the **tangent line** as the points get infinitesimally close. This is formalized as:

$$\text{If the limit } \lim_{{h \to 0}} \frac{f(c + h) – f(c)}{h} \text{ exists, the function is differentiable at point } c. $$

A classic example of non-**differentiability** is the absolute value function at x=0 due to a **sharp corner**.

In summary, a **differentiable** function will have a **graph** where a unique **tangent line** is definable at every point within its domain. If a function’s graph meets these criteria, it is safe to conclude that the function exhibits **differentiability**.

## Exploring Examples and Exceptions

When I consider if a function is **differentiable**, I first check if the **derivative** of the function exists.

With polynomials, it’s typically straightforward due to the **power rule**, which states that the **derivative** of $x^n$ is $nx^{n-1}$. So, if I have $f(x) = x^4 – 3x + 5$, the **derivative** $f'(x) = 4x^3 – 3$ exists everywhere, making the function **differentiable** throughout its domain.

However, I also watch out for exceptions. Consider the **absolute value function** $f(x) = |x|$, which isn’t **differentiable** at $x=0$.

The **left-hand limit** of the **derivative** as $x$ approaches zero from the left ($\lim_{x \to 0^-} \frac{|x|}{x}$) is $-1$, while the **right-hand limit** as $x$ approaches zero from the right ($\lim_{x \to 0^+} \frac{|x|}{x}$) is $1$. Since these two limits do not match, the **derivative** does not exist at $x=0$.

I also encounter **piecewise-defined** functions that require careful examination. For such functions, I check the **derivatives** of each piece separately and at the points where the pieces meet.

For example, given $f(x) = \begin{cases} x^2 + 2 & \text{for } x \leq 1 \ -2x + 5 & \text{for } x > 1 \end{cases}$, I must ensure the slopes of the two pieces are equal at $x=1$ for $f(x)$ to be **differentiable** at that point.

I sometimes come across functions with **vertical tangents** or where the **derivative** heads to infinity, which also indicates non-differentiability at those points. Lastly, there are some pathological cases like the **Weierstrass function**, which is continuous everywhere but **differentiable** nowhere, defying my intuition!

In summary, my journey through examples and exceptions sharpens my understanding of the concept of differentiability.

## Conclusion

In determining whether a function is **differentiable**, I always consider the core principle that a **differentiable** function must have a derivative at every point within its domain.

I remember that for **piecewise-defined functions**, continuity at the junction point (points where the piecewise segments meet) is essential — they need to have matching left-hand and right-hand limits for both the function itself and its first derivative.

When I come across a function like $f(x) = \left{ \begin{array}{lr} x^2 + 2 & : x \leq 1\ -2x + 5 & : x > 1 \end{array} \right.$, I check its **differentiability** at $x=1$ by comparing the limits of its derivatives from the left and the right.

Similarly, for **polynomials** or **transcendental functions**, I can affirm their **differentiability** by verifying the existence of their derivatives.

In essence, for a function to be considered **differentiable**, it must have a well-defined derivative across its entire domain.

The **existence** and **continuity of the derivative** are my guiding lights here. Armed with these concepts and by applying derivative rules, I can tackle the question of **differentiability** with confidence and precision.