2Finding derivatives of simple functionsEvery polynomial function (with domainR) is differentiable ateveryvalue ofx.To finish this subsection, here’s a proof of the sum rule, usingdifferentiation from first principles. It uses the Lagrange notation form ofthe sum rule, which is repeated below.Sum rule (Lagrange notation)Ifk(x) =f(x) +g(x), wherefandgare functions, thenk(x) =f(x) +g(x),for all values ofxat which bothfandgare differentiable.A proof of the sum ruleSuppose thatfandgare functions, and that the functionkis given byk(x) =f(x) +g(x). Letxdenote any value at which bothfandgaredifferentiable. To findk(x), you have to consider what happens to thedifference quotient forkatx, which isk(x+h)−k(x)h(wherehcan be either positive or negative, but not zero), ashgets closerand closer to zero. Sincek(x) =f(x) +g(x), this expression is equal to(f(x+h) +g(x+h))−(f(x) +g(x))h,that is,f(x+h) +g(x+h)−f(x)−g(x)h,which is equal tof(x+h)−f(x)h+g(x+h)−g(x)h.The expression in the first pair of large brackets is the difference quotientforfatx, and the expression in the second pair of large brackets is thedifference quotient forgatx. So ashgets closer and closer to zero, thevalues of these two expressions get closer and closer tof(x) andg(x),respectively. Hence the whole expression gets closer and closer tof(x) +g(x). Sok(x) =f(x) +g(x),which is the sum rule.249