What is Differentiability?
A function is differentiable at a point if it has a unique, well-defined derivative (slope) there. Geometrically, the graph must be smooth — no sharp corners or cusps.
A function f(x) is differentiable at x = a if the following limit exists and is finite:
h→0 ————————————
h
Left & Right Hand Derivatives
Just like limits, derivatives can be approached from both sides.
h→0⁺
RHD = f'(a⁺) = lim f(a+h) − f(a) / h
h→0⁺
💡 f(x) is differentiable at x = a if and only if LHD = RHD (both exist and are equal).
LHD = lim(h→0⁺) [|0−h| − |0|] / (−h) = lim h/(−h) = −1
RHD = lim(h→0⁺) [|0+h| − |0|] / h = lim h/h = +1
LHD ≠ RHD → NOT differentiable at x = 0 (sharp corner at origin)
💡 |x| is continuous at x = 0 but NOT differentiable — this shows continuity does NOT guarantee differentiability.
Relation: Continuity vs Differentiability
Differentiable ⟹ Continuous
But Continuous ⟹ Differentiable is NOT always true
✅ Continuous
- No breaks or holes
- LHL = RHL = f(a)
- Graph is unbroken
- May have sharp corners
✅ Differentiable
- Smooth, no sharp corners
- LHD = RHD exists
- Always continuous too
- Unique tangent at every point
⚠️ If NOT continuous → NOT differentiable. Discontinuity is a stronger condition — it immediately rules out differentiability.
Standard Derivatives (First Principles)
These are derived from the limit definition and must be memorised:
| f(x) | f'(x) = d/dx [f(x)] |
|---|---|
| xⁿ | nxⁿ⁻¹ |
| sin x | cos x |
| cos x | −sin x |
| tan x | sec²x |
| cot x | −cosec²x |
| sec x | sec x · tan x |
| cosec x | −cosec x · cot x |
| eˣ | eˣ |
| ln x | 1/x |
| constant c | 0 |
Rules of Differentiation
• Sum/Difference: d/dx [f ± g] = f' ± g'
• Product Rule: d/dx [fg] = f'g + fg'
• Quotient Rule: d/dx [f/g] = (f'g − fg') / g²
• Chain Rule: d/dx [f(g(x))] = f'(g(x)) · g'(x)
• Scalar: d/dx [kf] = k·f'
Outer function: sin(u), Inner: u = x²
d/dx[sin(x²)] = cos(x²) · d/dx[x²]
= cos(x²) · 2x = 2x cos(x²)
f = x², g = sin x → f' = 2x, g' = cos x
= f'g + fg' = 2x·sin x + x²·cos x
= 2x sin x + x² cos x