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The Derivatives of Polynomial Functions We have seen that derivatives of functions are of practical use because they represent instantaneous rates of change. If we have to compute derivatives using rst principles all the time then it is a very tedious and time- consuming process. Let's look at some rules that will make this process easier. 1. The Constant Function Rule If it) : k, where k is a constant, then f '(x) = 0. Since the graph of any constant function is a horizontal line with slope zero at each point, the derivative should be zero. 1e: f[x)=3 , f'[x]={} 2. The Power Rule If f(x} = x\" , where n is a real number, then f Tit) = nxn'l. In the previous sections investigation we discovered this. 1e: f(x]:x3 , thenf'[x]= j\"(i:)=.irS .then f'(x)= sixl=\\/; . six}= . six): . g'{Il= Proof: ( Note: n is a positive integer.) Using the definition of the derivative, f( x + 1) - f(x) f'(x) = lim , where f(x) = _x h-+0 (x + h )" -x" = lim (Factor) h ( x t h - x) [ ( x + h ) " - 1 + (x + h)" - 2 x + ... + (x+ h)x-2+ x -1] = lim = lim[(x + h)#-1 + (x+ 1)"-2x+ ... + (x+ h)x-2+x-1] (Divide out h) =m-1+ x-2 (x) + ... + (r)x-2+ x2-1 (Since there are n terms) = n-1\f3. The Constant Multiple Rule If f(x) =k . g(x) , where k is any constant, then f(x) = k .g'(x). Proof: Let f(x) = kg(x). By the definition of the derivative, f ( x + h) - f(x) f' (x) = lim h kg(x + h) - kg(x) = lim (Factor) 1-+0 h = lim k g( x + h) - 8(x) 1-0 h g( x + h) - g(x) = k lim (Property of limits) = kg' (x)Eg.2: Using the constant multiple rule. a) f(x) =12x3 then b ) f (x) =18x3 then f' ( x) = 12. 4 (x3) = 12. (3x2 ) = 36x2 f' (x) =18. d x3 dx dx = 18. (4x3 = 24x3 4. The Sum and Difference Rule If the functions p(x) and q(x) are differentiable, and f(x) = p(x)+q(x) then f '(x) = p'(x)+q'(x). Proof: Let f(x) = p(x) + q(x). By the definition of the derivative. f (x + h) - f(x) f'(x) = lim 1-0 h [p( x + h) + q(x + /1)] - [p(x) + q(x)] = lim 11-0 = lim [p(x + h) - p(x)] , [q(x+ h) - q(x)]] 1-+0 h h = lim ( [p(x + h) - p(x)]] + lim [ [q(x + 1) - q(x)]] h = p'(x) + q'(x)