Calculating derivatives ? a. ? or the following functions, find f using the de ? finition f (x) = lim f(x+hh?f(x). h?0 ? b. ?Determine an equation of the line tangent to the gr?ap?h?of? f at (?a, ?f(a ? )) ?for the given value of a. 2 f(x) = 3x+1 ;a=-1

SOLUTION STEP 1 (a). Given f(x) = 2 . 3x+1 We need to find the derivative of f(x) using the definition of the derivative The definition of the derivative is f (x) = lim f(x+h)f(x) h0 h 2 f(x+h) = 3(x+h)+1 2 2 2(3x+3h+1)(3x+1) Thus f (x) = lim 3x+3hh13x+1= lim h = lim h(3x+3h+1)(3x+1) h0 h0 h0 = lim 6 = 6 2 h0 (3x+3h+1)(3x+1) (3x+1) STEP 2 (b). Determine an equation of the line tangent to the graph of f at (a, f(a)) ,{ie,(-1,f(-1))}...