### 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-

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.

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

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