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We will start with the simplest type of differential, called an open differential. First well need to explore some terminology: The image below labels the components of an open differential. When a car is driving straight down the road, both drive wheels are spinning at the same speed. The input pinion is turning the ring gear and cage, and none of the pinions within the cage are rotating -- both side gears are effectively locked to the cage. Note that the input pinion is a smaller gear than the ring gear; this is the last gear reduction in the car. You may have heard terms like rear axle ratio or final drive ratio. These refer to the gear ratio in the differential. If the final drive ratio is 4.10, then the ring gear has 4.10 times as many teeth as the input pinion gear. See How Gears Work for more information on gear ratios. When a car makes a turn, the wheels must spin at different speeds. In the accompanying figure, you can see that the pinions in the cage start to spin as the car begins to turn, allowing the wheels to move at different speeds. The inside wheel spins slower than the cage, while the outside wheel spins faster. Need a tutor? Click this link and get your first session free! * AP® is a trademark registered and owned by the College Board, which was not involved in the production of, and does not endorse, this site. The following problems require the use of implicit differentiation. Implicit differentiation is nothing more than a special case of the well-known chain rule for derivatives. The majority of differentiation problems in first-year calculus involve functions y written EXPLICITLY as functions of x . For example, if , then the derivative of y is. However, some functions y are written IMPLICITLY as functions of x . A familiar example of this is the equation x2 + y2 = 25 , which represents a circle of radius five centered at the origin. Suppose that we wish to find the slope of the line tangent to the graph of this equation at the point (3, -4) . How could we find the derivative of y in this instance ? One way is to first write y explicitly as a function of x . Thus, x2 + y2 = 25 , y2 = 25 - x2 , and, where the positive square root represents the top semi-circle and the negative square root represents the bottom semi-circle. Since the point (3, -4) lies on the bottom semi-circle given by , the derivative of y is, i.e.,. Thus, the slope of the line tangent to the graph at the point (3, -4) is . Unfortunately, not every equation involving x and y can be solved explicitly for y . For the sake of illustration we will find the derivative of y WITHOUT writing y explicitly as a function of x . Recall that the derivative (D) of a function of x squared, (f(x))2 , can be found using the chain rule : . Since y symbolically represents a function of x, the derivative of y2 can be found in the same fashion :. Now begin with x2 + y2 = 25 . Differentiate both sides of the equation, getting D ( x2 + y2 ) = D ( 25 ) , D ( x2 ) + D ( y2 ) = D ( 25 ) , and 2x + 2 y y = 0 , so that 2 y y = - 2x , and, i.e.,. Thus, the slope of the line tangent to the graph at the point (3, -4) is . This second method illustrates the process of implicit differentiation. It is important to note that the derivative expression for explicit differentiation involves x only, while the derivative expression for implicit differentiation may involve BOTH x AND y . The following problems range in difficulty from average to challenging. PROBLEM 1 : Assume that y is a function of x . Find y = dydx for x3 + y3 = 4 . Click HERE to see a detailed solution to problem 1. PROBLEM 2 : Assume that y is a function of x . Find y = dydx for (x-y)2 = x + y - 1 . Click HERE to see a detailed solution to problem 2. PROBLEM 3 : Assume that y is a function of x . Find y = dydx for . Click HERE to see a detailed solution to problem 3. PROBLEM 4 : Assume that y is a function of x . Find y = dydx for y = x2 y3 + x3 y2 . Click HERE to see a detailed solution to problem 4. PROBLEM 5 : Assume that y is a function of x . Find y = dydx for exy = e4x - e5y . Click HERE to see a detailed solution to problem 5. PROBLEM 6 : Assume that y is a function of x . Find y = dydx for . Click HERE to see a detailed solution to problem 6. PROBLEM 7 : Assume that y is a function of x . Find y = dydx for . Click HERE to see a detailed solution to problem 7. PROBLEM 8 : Assume that y is a function of x . Find y = dydx for . Click HERE to see a detailed solution to problem 8. PROBLEM 9 : Assume that y is a function of x . Find y = dydx for . Click HERE to see a detailed solution to problem 9. PROBLEM 10 : Find an equation of the line tangent to the graph of (x2+y2)3 = 8x2y2 at the point (-1, 1) . Click HERE to see a detailed solution to problem 10. PROBLEM 11 : Find an equation of the line tangent to the graph of x2 + (y-x)3 = 9 at x=1 . Click HERE to see a detailed solution to problem 11. PROBLEM 12 : Find the slope and concavity of the graph of x2y + y4 = 4 + 2x at the point (-1, 1) . Click HERE to see a detailed solution to problem 12. PROBLEM 13 : Consider the equation x2 + xy + y2 = 1 . Find equations for y and y in terms of x and y only. Click HERE to see a detailed solution to problem 13. PROBLEM 14 : Find all points (x, y) on the graph of x23 + y23 = 8 (See diagram.) where lines tangent to the graph at (x, y) have slope -1 . Click HERE to see a detailed solution to problem 14. PROBLEM 15 : The graph of x2 - xy + y2 = 3 is a tilted ellipse (See diagram.). Among all points (x, y) on this graph, find the largest and smallest values of y . Among all points (x, y) on this graph, find the largest and smallest values of x . Click HERE to see a detailed solution to problem 15. PROBLEM 16 : Find all points (x, y) on the graph of (x2+y2)2 = 2x2-2y2 (See diagram.) where y = 0. Click HERE to see a detailed solution to problem 16. Click HERE to return to the original list of various types of calculus problems. Your comments and suggestions are welcome. Please e-mail any correspondence to Duane Kouba by clicking on the following address : Duane Kouba1998-06-23
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