Differentiation Notes Definition, Basic Rules of Derivation

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. For Example- See graph of f(x) = |x|
If the graph of the function exhibits a sharp corner at x=a, it means unit tangent can’t be drawn at x = a.

Differentiability in an Interval

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.

Q-> Try to solve these functions?

(i) f(x) = |x³| (ii) f(x) = |x(x-1)|(iii) f(x) = (x-1)|x² – 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)= x²sin(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.

Methods of Differentiation

Symbols used for derivative

dy/dx or f ‘(x) or D(y) or y’ or y₁→ First Order Derivative

d²y/dx² or f ”(x) or D²(y) or y” or y₂→ Second Order Derivative

Derivative of some standard Functions

d(xⁿ)/dx =nx^n-1 n ∈ R; x>0
d(a^x)/dx =a^xln a a>0
d(e^x)/dx =e^x
d(ln x)/dx = 1/x

Trigonometric Functions

d(sin x)/dx = cos x
d(cos x)/dx = -sin x
d(sec x)/dx = sec x tan x
d(cosec x)/dx = -cot x cosec x
d(tan x)/dx = sec²x
d(cot x)/dx = –cosec²x

Inverse Trigonometric Functions

d(sin^-1x)/dx = 1/(1-x²)^1/2 -1< x <1
d(cos^-1x)/dx = -1/(1-x²)^1/2 -1< x <1
d(cot^-1x)/dx = -1/(1+x²) x∈ R
d(tan^-1x)/dx = 1/(1+x²) x∈ R
d(sec^-1x)/dx = 1/|x|(x²-1)^1/2|x|>1
d(cosec^-1x)/dx = -1/|x|(x²-1)^1/2 |x|>1

Rules in Differentiation

Sum or Difference Rule

If h (x) = f(x) ± g (x)

then h’ (x) = f ‘(x) ± g’ (x)

Constant Multiple Rule

If h (x) = k f(x) where k is any constant h’ (x) = k f ‘(x)

Product Rule

h (x) = f(x) × g (x) then h’ (x) = f ‘(x) g (x) + g’ (x) f(x) Note- (fgh)’ = (fg)h’ + (fh)g’ + (gh)f ‘ or (fhg)’ = {f(gh)’ + g(fh)’ + h(fg)’}/2

Quotient Rule

d{f(x)/g(x)}/dx = {f ‘(x) g (x) + g’ (x) f(x)}/g²(x) Also- d{1/f (x)}/dx = – f ‘(x)/f²(x)

Chain Rule

If y = f(t); t = g(u); u = h(x) then, dy/dx = dy/dt × dt/du × du/dx = f ‘(t) × g'(u) × h'(x)

Physical Meaning of the term Derivative

Geometrical Meaning of the term Derivative