## Differentiability in an Interval

Contents

- A function is said to be differentiable in (a, b) if it is differentiable at each and every point of the interval.
- Differentiable at [a, b] if

It is differentiable in (a, b)

f’ (a^{+}) = limit(h→0) {(f(a+h)- f(a)) / h} = a finite quantity

f’ (a^{–}) = limit(h→0) {(f(b-h)- f(a)) /- h} = a finite quantity

- All trigonometric, logarithmic, exponential and polynomial functions are differentiable in their domain (Inverse Trigonometric Functions are not included).

**Note-**

- Derivative in an interval should be checked at all those points where f(x) may be discontinuous.
- For |f(x)| differentiability should be checked at those points where f(x) = 0.

*For Example-*

Consider a differentiable function f(x) whose graph is given as- Fig.- A

|f(x)| → Fig.- B

F(x) = 0 at A, B, C

At x= A |f(x)| is derivable; but may or may not be derivable at x = B & C.

**Read More- Basic Meaning of Differentiations**

**Q-> Try to solve these functions?**

(i) f(x) = |x^{3}| (ii) f(x) = |x(x-1)|(iii) f(x) = (x-1)|x^{2} – 3x + 2|

**Important Points to Remember**

- If f(x) and g(x) both are differentiable function at x = a then f(x) ± g(x), f(x)×g(x) and f(x)/g(x)[g(x)≠0] will also be differentiable at x = a.
- If f(x) differentiable and g(x) is not differentiable at x = a then f(x) ± g(x) will not be differentiable but nothing definite can be said about derivative of function f(x) × g(x) and f(x)/g(x)[where g(x) ≠ 0] at x = a.
- If f(x) and g(x) both are not differentiable at x = a then nothing definite can be said about derivative of function f(x) × g(x) and f(x)/g(x) [where g(x)≠0] at x = a.
- A derivative of a continuous function need not be continuous.

Ex- f(x)= Two cases

Case-1 f(x)= x^{2}sin(1/x); x≠0

Case-2 f(x)= 0; x=0

- If f(x) is derivable at x = a and f(a)=0 and g(x) is continuous at x = a then product function f(x)g(x) will also be derivable at x =a.
- If f(x) is differentiable at x = a and p(h) and g(h) approaches to zero as h → 0

limit (h→0) f(a+p(h)) – f(a+ g(h))/ p(h)- g(h) = f’(a)

**Determination of functions which are differentiable and satisfying some given functional rule**

**Four Basic Steps**

- Write the expression for f'(x) whether LHD or RHD

f'(x) = limit(h→0) {f(x+h) – f(x)}/ h

- Manipulate f(x+h)-f(x) using functional rule and get f'(x) in its simplifying form.
- Integrate both sides and f(x) with the constant of its integration.
- Using a suitable value of ‘x’, get the value of the constant of integration.