# Differentiability of a Function at a Point

### Definition:

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

### Conditions for Differentiability:

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

2. Continuity at the Point: If a function is differentiable at $c$, it must also be continuous at $c$.

### Example:

Consider the function $f\left(x\right)={x}^{2}$.

Let's investigate the differentiability of $f\left(x\right)$ at $x=2$.

### Step 1: Calculating the Derivative:

The derivative of $f\left(x\right)={x}^{2}$ is ${f}^{\mathrm{\prime }}\left(x\right)=2x$.

### Step 2: Evaluating ${f}^{\mathrm{\prime }}\left(2\right)$:

Substituting $x=2$ into the derivative ${f}^{\mathrm{\prime }}\left(x\right)=2x$:

${f}^{\mathrm{\prime }}\left(2\right)=2×2=4$

### Step 3: Conclusion:

The derivative ${f}^{\mathrm{\prime }}\left(2\right)=4$ exists, indicating that $f\left(x\right)={x}^{2}$ 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 well-defined 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.

• Real-world Applications: Applied fields use differentiability to model various phenomena, such as motion, growth, and economics.