Review of Derivatives Power rule, product rule, quotient rule and chain rule The Power Rule Remember, the power rule only works on functions of the form y = xn. The power rule says that y’ = nxn-1 Examples: y = x2, so y’ = 2x y =x1/2, so y’ = ½x-1/2 y = x -1, so y’ = -x -2 The Product Rule The product rule can be used when two functions are multiplied together. If y = f(x)g(x), then y’ = f’(x)g(x) + f(x)g’(x) Examples: If y = xsinx, then y’ = sinx +xcosx If y = (3x)(5x+1), then y’ = (3)(5x+1) + (3x)(5) Of course, you must remember to simplify your answers! The Quotient Rule The quotient rule can be used when two functions are being divided. If y = f(x)/g(x), then y’ = [g(x)f’(x) – f(x)g’(x)]/(g(x))2, or (lodhi – hidlo) / lolo ! Example: If y = sinx/cosx, then y’ = [cosx(cosx) – sinx(sinx)]/cos2x What does this simplify to??? Trigonometric Derivatives If If If If If If y y y y y y = = = = = = sinx, then y’ = cosx cosx, then y’ = -sinx tanx, then y’ = sec2x secx, then y’ = secxtanx cscx, then y’ = -cscxcotx cotx, then y’= -csc2x The Chain Rule The chain rule is used on composition functions. You must identify the inside function and the outside function. The chain rule says if y = f(g(x), then y’ = f’(g(x))*g’(x), or the derivative of the inside times the derivative of the outside The Chain Rule (cont’d) Examples: If y = sin(x2), then y’ = 2xcos(x2) If y = (2x+1)3 then y’ = 2*3(2x+1)2 Remember, the product rule and the quotient rule may also need to be used along with the chain rule!! If y = (2x+1)3(3x+2)2, then y’ = 2*3(2x+1)2(3x+2)2 + (2x+1)3(3)(2(3x+2)) Don’t forget to simplify!!!