Physical Meaning of the term Derivative
Instantaneous rate of change of function w.r.t independent variable.
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-
mAB = 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
mAB’ = 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.
- Differentiability ⇒ Continuity
- Continuity ⁄⇒ Differentiability
- Discontinuity ⇒ Non-Differentiability
- Non-Differentiability ⁄⇒ Discontinuity
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.