Derivative of parametric functions: Survey of methods

Consider a parametric function y = y(x) given by a parametric curve
x = x(t),
y = y(t)
on a neighborhood of some point (x₀,y₀) corresponding to the time t₀. Assume that the functions x(t), y(t) are differentiable at t₀.

We find the derivative y′ with respect to x by

If the functions x(t), y(t) are actually twice differentiable at t₀, then we also have the second derivative

Example: We have shown in Solved Problems in Functions - Solved Problems - Implicit and parametric functions that the parametric curve given by
x = e^t − 1,
y = e^2t − 2e^t
is actually a part of the parabola, in particular it can be described as a function y(x) on a neighborhood of the point (0,−1). Find y′(0), the derivative with respect to x at 0.

Solution: We use the above formula. The time that corresponds to the point (0,1) is t = 0.

By the way, this shows that there is a horizontal tangent line at that point, which fits with the picture we obtained before.

Remark: The tangent vector to a parametric curve   x = x(t), y = y(t)   can be found as   v = (,).

Graphing parametric functions
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