Differentiability of a Function at a Point
Definition:
 Differentiability at a Point $c$: A function $f(x)$ is said to be differentiable at a point $c$ in its domain if the derivative ${f}^{\mathrm{\prime}}(c)$ exists.
Conditions for Differentiability:

Existence of the Derivative: The limit ${\mathrm{lim}}_{h\to 0}\frac{f(c+h)f(c)}{h}$ must exist for ${f}^{\mathrm{\prime}}(c)$ to exist.

Continuity at the Point: If a function is differentiable at $c$, it must also be continuous at $c$.
Example:
Consider the function $f(x)={x}^{2}$.
Let's investigate the differentiability of $f(x)$ at $x=2$.
Step 1: Calculating the Derivative:
The derivative of $f(x)={x}^{2}$ is ${f}^{\mathrm{\prime}}(x)=2x$.
Step 2: Evaluating ${f}^{\mathrm{\prime}}(2)$:
Substituting $x=2$ into the derivative ${f}^{\mathrm{\prime}}(x)=2x$:
${f}^{\mathrm{\prime}}(2)=2\times 2=4$
Step 3: Conclusion:
The derivative ${f}^{\mathrm{\prime}}(2)=4$ exists, indicating that $f(x)={x}^{2}$ is differentiable at $x=2$.
Understanding Differentiability:

Derivative Existence: If the derivative of a function exists at a specific point, the function is differentiable at that point.

Tangent Line Interpretation: Differentiability implies the existence of a welldefined tangent line to the curve at that point.
Importance of Differentiability:

Rate of Change: Differentiability helps in understanding rates of change and slopes of functions.

Realworld Applications: Applied fields use differentiability to model various phenomena, such as motion, growth, and economics.