Derivative of the Square Root Function a) Use implicit diﬀerentiation to ﬁnd the derivative of the inverse of f (x) = x2 for x > 0. b) Check your work by ﬁnding the inverse explicitly and then taking its deriva tive. Solution If you’re having trouble with this problem, it may help to review Professor Jerison’s example of the derivative of the arctangent function. a) Use implicit diﬀerentiation to ﬁnd the derivative of the inverse of f (x) = x2 for x > 0. dy We wish to ﬁnd y � = where y = f −1 (x) and f (x) = x2 . Our goal is dx to practice using implicit diﬀerentiation, so instead of ﬁnding f −1 (x) right away we will follow the steps outlined in lecture. We start with our statement y = f −1 (x) and apply the function f to both sides: y = f −1 (x) f (y) = f (f −1 (x)) y2 = x Next, we diﬀerentiate both sides of the equation then solve for y � . d 2 y dx 2yy � = y� = = d x dx 1 1 2y This gives us an expression describing y � in terms of y. We return to our equation y 2 = x to rewrite it in terms of x. Remembering that x > 0, we ﬁnd: √ y = x. Plugging this in to our equation for y � we get: y� = 1 1 = √ . 2y 2 x 1 We conclude that if f (x) = x2 , the derivative of f −1 is √ . 2 x b) Check your work by ﬁnding the inverse explicitly and then taking its deriva tive. √ If f (x) = x2 then f −1 (x) = x for x > 0. d √ x dx = = = d 1/2 x dx 1 −1/2 x 2 1 √ . 2 x