Parametrics – Curves, Calculus, Polar Coordinates (continued)
Application of the chain
rule gives dy/dt = dy/dx dx/dt Then the first derivative
is dy/dx = [dy/dt] / [dx/dt] provided
that dx/dt ≠ 0. Higher derivatives may also be calculated for parametric
representation of functions. For the second
derivative d2y/dx2 = d/dx [dy/dx]. So replace y with
dy/dx in dy/dx = [dy/dt] / [dx/dt] This substitution yields d2y/dx2 = d/dt [dy/dx]
/ [dx/dt] for the second derivative. Click here for examples of
derivatives of a function in parametric form. |
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Sometimes problems in
calculus may be more easily represented using a coordinate system other than cartesian coordinates (x,y). One such system is the polar coordinate system. The figure below shows the graphical
relation between cartesian coordinates (x,y) and polar coordinates (r, θ) . Here we see that x
= r cos
θ and y
= r sin θ Click here for examples
involving polar coordinates. |