# Differentiability

## Differentiability

### Definition:

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

### Conditions for Differentiability:

1. Existence of Derivative: The limit ${\mathrm{lim}}_{h\to 0}\frac{f\left(c+h\right)-f\left(c\right)}{h}$ exists.

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

### Graphical Interpretation:

• Tangent Line: A function is differentiable at a point if there exists a unique tangent line to the curve at that point.

• Smooth Curve: Differentiability implies a smooth, non-cornered curve without abrupt changes.

### Differentiability Rules:

1. Sum/Difference Rule: $\left(f±g{\right)}^{\mathrm{\prime }}\left(x\right)={f}^{\mathrm{\prime }}\left(x\right)±{g}^{\mathrm{\prime }}\left(x\right)$

2. Product Rule: $\left(f\cdot g{\right)}^{\mathrm{\prime }}\left(x\right)={f}^{\mathrm{\prime }}\left(x\right)\cdot g\left(x\right)+f\left(x\right)\cdot {g}^{\mathrm{\prime }}\left(x\right)$

3. Quotient Rule: ${\left(\frac{f}{g}\right)}^{\mathrm{\prime }}\left(x\right)=\frac{{f}^{\mathrm{\prime }}\left(x\right)\cdot g\left(x\right)-f\left(x\right)\cdot {g}^{\mathrm{\prime }}\left(x\right)}{\left(g\left(x\right){\right)}^{2}}$

4. Chain Rule: $\left(f\left(g\left(x\right)\right){\right)}^{\mathrm{\prime }}={f}^{\mathrm{\prime }}\left(g\left(x\right)\right)\cdot {g}^{\mathrm{\prime }}\left(x\right)$

### Understanding Differentiability:

• Smoothness of Function: Differentiability implies the function is smooth without abrupt changes.

• Slope at a Point: Differentiability indicates the existence of a well-defined slope at a specific point on the curve.

### Testing Differentiability:

1. Using Derivative Definition: Calculating the derivative using the definition of the derivative.

2. Applying Differentiability Rules: Employing derivative rules to find the derivative of a function.