Revision Notes - Differentiability Notes | Study JEE Main & Advanced Mock Test Series - JEE
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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^{+}) = t_{1} (a finite number) and f'(a^{-}) = t_{2} (a finite number) and if t_{1} ≠ 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) ≠ 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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