Differentiability

The derivative of f, denoted by f'(x) is given by f'(x) = lim_?x→0 (? y)/(? x) = dy/dx
The right hand derivative of f at x = a is denoted by f'(a⁺) and is given by f'(a⁺) = lim_h→0⁺(f(a+h)-f(a))/h
The left hand derivative of f at x = a is denoted by f'(a^-) and is given by f'(a^-) = lim_h→0^- (f(a-h)-f(a))/-h
For a function to be differentiable at x=a, we should have f'(a^-)=f'(a⁺) i.e. lim_h→0 (f (a-h)-f (a))/ (-h) = lim_h→0 (f (a+h)-f (a))/h.
lim_h→0 sin 1/h fluctuates between -1 and 1.
If at a particular point say x =a, we have f'(a⁺) = t₁(a finite number) and f'(a^-) = t₂ (a finite number) and if t₁ ≠ t₂, then f' (a) does not exist, but f(x) is a continuous function at x = a.
Continuity and differentiability are quite interrelated. Differentiability alwaysimpliescontinuity but the converse is not true. This means that a differentiable function is always continuous but if a function is continuous it may or may not be differentiable.

Some basic formulae:

If a function is not derivable at a point, it need not imply that it is discontinuous at that point. But, however, discontinuity at a point necessarily implies non-derivability.
In case, a function is not differentiable but is continuous at a particular point say x = a, then it geometrically implies a sharp corner at x = a.
A function f is said to be derivable over a closed interval [a, b] if :

1.For the points a and b, f'(a⁺) and f'(b^-) exist and

2.For ant point c such that a < c < b, f'(c⁺)and f'(c^-) exist and are equal.

If y = f(u) and u = g(x), then dy/dx = dy/du.du/dx = f'(g(x)) g'(x). This method is also termed as the chain rule.
For composite functions, differentiation is carried out in this way:

If y = [f(x)]ⁿ, then we put u = f(x). So that y = uⁿ. Then by chain rule:

dy/dx = dy/du.du/dx = nu^(n-1)f' (x) = [f(x)]^(n-1) f' (x)

If x and y are functions of parameter t, first find dx/dt and dy/dt separately. Then dy/dx=(dy/dt)/(dx/dt).

If the functions f(x) and g(x) are derivable at x = a, then the following functions are also derivable:

1.f(x) + g(x)

2.f(x) - g(x)

3.f(x) . g(x)

4.f(x) / g(x), provided g(a) ≠ 0

If the function f(x) is differentiable at x =a while g(x) is not derivable at x = a, then the product function f(x). g(x) can still be differentiable at x = a.
Even if both the functions f(x) and g(x) are not differentiable at x = a, the product function f(x).g(x) can still be differentiable at x = a.
Even if both the functions f(x) and g(x) are not derivable at x = a, the sum function f(x) + g(x) can still be differentiable at x = a.

If function f(x) is derivable at x = a, this need not imply that f'(x) is continuous at x = a.