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28, , Differentiation, , , , , , DERIVATIVE OF A FUNCTION, , Let y = f(x) be a given continuous function. Then, the value of y depends upon the, value of x and it changes with a change in the value of x. We use the word, increment to denote a small change, i.e., an increase or decrease, in the values of x, and y., , Let dy be an increment in y, corresponding to an increment dx in x. Then,, , y = f(x) and y + dy = f(x + dx)., On subtraction, we get dy = f(x + dx) — f(x)., By _ f(x + 8x) ~ FO),, , bx 8x, So, dim Y= tim +82) — FO), br>0 6x br 0 6x, , The above limit, if it exists finitely, is called the derivative or differential, coefficient of y = f(x) with respect to x, and it is denoted by, , dy gd, i Ze {f(x)} or f(x)., , The process of finding the derivative is known as differentiation., , remark 2 = tim &Y = tim f+ 8) - fC),, dx 8830 6x 8x50 6x, DERIVATIVE ATA POINT The value of f’(x), obtained by putting x =a, is called the, derivative of f(x) at x =a, and it is denoted by f’(a) or (eI, XJ yea, , If f’(a) exists, we say that f(x) is differentiable at x =a., , If f’(x) exists for every value of x in the domain of the function, we say, that f(x) is differentiable., GEOMETRICAL SIGNIFICANCE OF A DERIVATIVE, Let y=f(x) be a continuous function., Then, we draw its graph. Suppose it is a, curve. Let a be a point in the domain of the, given function. Let Pla, f(a)] be a point on, this curve, and let Qla+h, f(a+h)] be, some neighbouring point on it. x, , Slope of chord PQ = fa+h) -f@, , SOBAN RANA “*"?, , 838, , , , Q{(ath), flath)]

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Differentiation 839, , _f¢ (a + hi) — f(a) |, h, , Let the point Q move along the curve such that Q > P., , As Q comes closer and closer to P along the curve, the chord QP, approaches the tangent at P., , This happens when h > 0., , . f(at+h)-f(a)_,., lim —— 4 jim (slope of chord PQ), , or f’(a) = the slope of the tangent at P., , Hence, the derivative of f(x) at x =a is the slope of the tangent to the curve, y = f(x) at the point [a, f(a)]., PHYSICAL SIGNIFICANCE OF A DERIVATIVE Lets = f(t) be a function representing, the distance travelled by a particle moving in a straight line in time t., , Let (s + 5s) be the distance travelled by it in time (t + 5t)., , . s+s=f(t + 5t)., , On subtraction, we get ds = f(t + 5t) — f(é)., , 8s _ ft+ 84) -f(), st 3t ,, , Now, when 6f— 0, the average velocity becomes the instantaneous, velocity., , average velocity =, , velocity at time f = lim a = lim fe so 7H. =F)., &o0St 0 ot, , DIFFERENTIATION FROM THE FIRST PRINCIPLE Obtaining the derivative of a given, function by using the definition is called differentiation from the first principle or, ab initio or by delta method., , Some Important Derivatives Using the First Principle, , THEOREM 1 From the first principle, we have, , n-1, , (x") =nx"~", where nis a fixed number, integer or rational., , dx, , PROOF Let y=x". --- (i), Let dy be an increment in y, corresponding to an increment 6x in x., Then, y + dy =(x + 6x)". .» (ii), On subtracting (i) from (ii), we get dy =(x + dx)" — x", , dy _ (x + 8x)" — x", , 8x bx, , ay = lim oy, , or, , SOBAN RANA“ "=