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- The derivative of f, denoted by f'(x) is given by f'(x) = lim
_{?}_{xâ†’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_{hâ†’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_{hâ†’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_{hâ†’0}(f (a-h)-f (a))/ (-h) = lim_{hâ†’0}(f (a+h)-f (a))/h. - lim
_{hâ†’0}sin 1/h fluctuates between -1 and 1. - If at a particular point say x =a, we have f'(a
^{+}) = t_{1}(a finite number) and f'(a^{-}) = t_{2}(a finite number) and if t_{1}â‰ t_{2}, 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 = u^{n}. Then by chain rule:

dy/dx = dy/du.du/dx = nu^{(n-1)}f' (x) = [f(x)]^{(n-1)}f' (x) **Differential calculus**problems involving parametric functions:

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) â‰ 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.

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