A derivative formula d 2 a. Use the definition of the derivative to determine dx(ax +bx+c) , ?wh ? ere? ,? , ? and ?c are constants. d 2 b. Use the result of part (a) to find dx(4x ?3x+10) , .

Solution 33E STEP 1 (a). 2 Here f(x) = ax + bx + c We have to find d f(x) = f(x) dx f(x+h)f(x) The definition of f (x) = lim h h0 Thus on substituting the given values into the definition,we get 2 2 2 2 2 f (x) = lim (a(x+h) +b(x+h)+c)(ax +bx+c= lim (a(x +h +2xh)+bx+bh+c)(ax +bx+c = h0 h h0 h (ax +ah +2axh+bx+bh+c)(ax +bx+c ) (ah +2axh+bh) lim h = lim h = lim ah + 2ax + b h0 h0 h0 Therefore f (x = 2ax + b STEP 2 (b). d 2 Here,using the above result ,we need to find the value of dx(4x 3x + 10) d (ax + bx + c) = 2ax + b dx d 2 Therefore, dx(4x 3x + 10) = 2 × 4x 3 = 8x 3 f (4x 3x + 10) = 8x 3