Problem 1.6.12. (Implicit Differentiation). You know how to find the derivative dy/dx when y(x) is given. Suppose instead I tell you that y and x are related by an equation, say x2 + y² = R² and ask you to find the derivative at each point. There are two ways. The first is to solve for y as a function of x and then let your spinal column take over, i.e., by changing x infinitesimally and computing the corresponding change in y given by the functional relation. The second is to imagine changing x and y infinitesimally while preserving the constraining relation (a circle in our example). The latter condition allows us to relate the infinitesimals Ar and Ay and allows us to compute their ratio in the usual limit. Show that the derivative computed this way agrees with the first method. Find the slope at the point (2, 3) on the ellipse 3x2 + 4y2 = 48 using implicit differentiation. %3D %3D
Problem 1.6.12. (Implicit Differentiation). You know how to find the derivative dy/dx when y(x) is given. Suppose instead I tell you that y and x are related by an equation, say x2 + y² = R² and ask you to find the derivative at each point. There are two ways. The first is to solve for y as a function of x and then let your spinal column take over, i.e., by changing x infinitesimally and computing the corresponding change in y given by the functional relation. The second is to imagine changing x and y infinitesimally while preserving the constraining relation (a circle in our example). The latter condition allows us to relate the infinitesimals Ar and Ay and allows us to compute their ratio in the usual limit. Show that the derivative computed this way agrees with the first method. Find the slope at the point (2, 3) on the ellipse 3x2 + 4y2 = 48 using implicit differentiation. %3D %3D
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Problem 1.6.12. (Implicit Differentiation). You know how to find the derivative
dy/dx when y(x) is given. Suppose instead I tell you that y and x are related by
an equation, say x2 + y² = R² and ask you to find the derivative at each point.
There are two ways. The first is to solve for y as a function of x and then let
your spinal column take over, i.e., by changing x infinitesimally and computing
the corresponding change in y given by the functional relation. The second is to
imagine changing x and y infinitesimally while preserving the constraining relation
(a circle in our example). The latter condition allows us to relate the infinitesimals
Ar and Ay and allows us to compute their ratio in the usual limit. Show that the
derivative computed this way agrees with the first method.
Find the slope at the point (2, 3) on the ellipse 3x2 + 4y2 = 48 using implicit
differentiation.
%3D
%3D
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