3.7 Implicit Differentiation Niagara Falls, NY & Canada Photo by Vickie Kelly, 2003 Greg Kelly, Hanford High School, Richland, Washington Before we start, we are going to load the Calculus Tools flash application software to your calculator. 1. Connect the sending and receiving calculators together. 2. On the sending unit, press F7 . then 2nd Press F4 2nd VAR-LINK , and to select Calculus Tools . 3. On the receiving unit, press 4. On both units, press F3 4. On the receiving unit, select 5. On the sending unit, select 2nd VAR-LINK . (Link). 2 (Receive). 1 (Send). This is not a function, but it would still be nice to be able to find the slope. x2 y 2 1 d 2 d 2 d x y 1 dx dx dx Do the same thing to both sides. Note use of chain rule. dy 2x 2 y 0 dx dy 2y 2 x dx dy 2 x dx 2 y dy x dx y 2 y x 2 sin y d d 2 d 2y x sin y dx dx dx This can’t be solved for y. dy 2x dx 2 cos y dy dy 2 2 x cos y dx dx dy dy 2 cos y 2x dx dx This technique is called implicit differentiation. 1 Differentiate both sides w.r.t. x. dy 2 cos y 2 x dx dy 2 Solve for . dx Find the equations of the lines tangent and normal to the curve x 2 xy y 2 7 at (1, 2) . We need the slope. Since we can’t solve for y, we use dy implicit differentiation to solve for . dx x 2 xy y 2 7 Note product rule. dy dy dy 2x x y 2 y 0 dx dx dy dy 2x x y 2 y 0 dx dx dy 2 y x y 2x dx y 2x dx 2 y x 2 2 1 22 4 m 2 2 1 4 1 5 Find the equations of the lines tangent and normal to the curve 4 m 5 x 2 xy y 2 7 at (1, 2) . tangent: normal: 4 y 2 x 1 5 5 y 2 x 1 4 4 4 y2 x 5 5 5 5 y2 x 4 4 4 14 y x 5 5 5 3 y x 4 4 Higher Order Derivatives d2y 3 2 Find if 2 x 3 y 7 . 2 dx 2 y 2 x x y 3 2 y 2x 3 y 7 y2 6 x 6 y y 0 2 2x x 2 y 2 y y y 6 y y 6 x 2 6 x 2 y 6 y 2 x y y 2x x 2 x 2 y 2 y y y 2x x 4 y 3 y y Substitute y back into the equation. Now let’s do one on the TI-89: 2 d y 3 2 2 x 3 y 7 Find . 2 dx APPS Select Calculus Tools and press ENTER . If you see this screen, press , change the mode settings as necessary, and ENTER press again. APPS Now let’s do one on the TI-89: 2 d y 3 2 2 x 3 y 7 Find . 2 dx APPS Press Select Calculus Tools and press Press F2 4 (Deriv) (Implicit Difn) Enter the equation. (You may have to unlock the alpha mode.) Set the order to 2. Press ENTER . ENTER . Press ESC and then HOME to return your calculator to normal. p