**Physical Meaning of the term Derivative**

Instantaneous rate of change of function w.r.t independent variable.

limit (Δy/Δx)

Δx→0

**Geometrical Meaning of the term Derivative**

It gives the slope of the tangent drawn to the curve at x=a.

**Tangent- **It is the limiting case of secant slope of the secant.

**Slope of Secant AB-**

m_{AB} = f(a+h) – f(a) ⁄ (a + h – a)

f ’ (a^{+}) = lim (h→0) f(a+h) – f(a) / h = slope of tangent at A from RHS

f ’ (a^{+}) = Right hand derivative at x=a

m_{AB’} = f(a-h) – f(a) ⁄ (a – h – a)

f ’ (a^{–}) = lim (h→0) f(a+h) – f(a) / -h = slope of tangent at A from LHS

f ’ (a^{–}) = Left hand derivative at x=a

If f ’ (a^{+}) = f ’ (a^{–}) = a finite quantity, then f(x) is said to be differentiable at x=a; and in this case curve y = f(x) has a unique tangent (Not Shown) of finite slope at x = a.

f ’ (a^{+}) = f ’ (a^{–}) = f ’ (a) = finite slope of the tangent at x= 0.

**THEOREM-** If a function is differentiable at x=a then it is also continuous at x= a.

**NOTE- **

- Differentiability ⇒ Continuity
- Continuity ⁄⇒ Differentiability
- Discontinuity ⇒ Non-Differentiability
- Non-Differentiability ⁄⇒ Discontinuity

**NOTE- **

If f ’ (a^{+}) = p and f ’ (a^{–}) = q where ‘p’ and q” are finite values, then

- If p = q then f(x) is differentiable at x=a and continuous at x=a.
- If p ≠ q then f(x) is not differentiable at x=a but may or may not be continuous at x=a.
- f(x) is not differentiable but continuous at x=a then the geometrical graph of the function has a sharp corner at x=a. Ex- See graph of f(x) = |x|
- If the graph of the function exhibits a sharp corner at x=a, means unit tangent can’t be drawn at x=a.

*Read More- Differentiability in an Interval & Imp. Points*