Module 23  Parametric Equations  
Introduction  Lesson 1  Lesson 2  Lesson 3  SelfTest  
Lesson 23.2: Chain Rule for Parametric Equations  
This lesson investigates the procedure to find derivatives, such as and , for parametric equations x = f(t), y = g(t). The Chain Rule Suppose a curve is defined by the parametric equations x = f(t) y = g(t) The Chain Rule states that the derivative on the parametric curve is the ratio of to . Higher derivatives are found in a similar fashion. Symbolically,
Finding First and Second Derivatives The procedure below will create two functions that return the values of and for parametric equations defined by x(t) and y(t). You can then define x(t) and y(t) and use the new functions to find these two derivatives for the resulting parametric curve.
Use the new functions to find dy/dx and d^{2}y/dx^{2} for the parametric equations x(t) = sec t y(t) = tan t Define the parametric equations below. Recall that sec
Find and by using the functions defined earlier.
Finding Derivatives from a Graph The slope of the tangent line at a point on the graph of a parametric curve can be found by using the "Derivative" feature of the Math menu on the Graph screen.
The graph is a hyperbola and the diagonal lines are not a part of the graph. The two diagonal lines are similar to the vertical lines that often appear in graphs of functions that have vertical asymptotes.
You can use the "Derivatives" feature in the F5:Math menu on the Graph screen to find the value of at t = 0.5.
23.2.1 Find the value of at t = 0.5 by entering dydx(0.5) in the Edit Line of the Home screen and compare it to the value found using the "Derivative" feature of the Graph screen. Click here for the answer. Finding the Equation of the Tangent Line
The tangent to the curve at the point when t = 0.5 is drawn and the equation of the tangent line is shown at the bottom of the screen. 23.2.2 Find the slope of the tangent line when t = 2 for the parametric equations .
Click here for the answer. 23.2.3 Display the curve and find the equation of the tangent line to the curve when t = 2.Click here for the answer. 

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