## Section 4.1 | Derivatives of Parametric Curves

**THEOREM | DERIVATIVES OF PARAMETRIC-DEFINED CURVES**

Let \(f\) and \(g\) be differentiable functions, where \(x = f(t)\) and \(y = g(t)\). Then \[\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}, \quad \text{if } \frac{dx}{dt} \neq 0.\]Moreover,\[\frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right) = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}.\]

Notice that the parametric curve has a horizontal tangent when \(\frac{dy}{dt} = 0\) (provided \(\frac{dx}{dt} \neq 0\)) and a vertical tangent when \(\frac{dx}{dt} = 0\) (again, provided \(\frac{dy}{dt} \neq 0\)).

**EXAMPLE 1 [P.1] | Finding the equation of a line tangent to a parametric curve**

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. \[x(t) = t \cos{(t)}, \quad y(t) = t \sin{(t)}; \quad t = \pi.\]

**EXAMPLE 2 [P.2] | Determining concavity**

Find \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\) for \[x(t) = \cos{(2t)}, \quad y(t) = \cos{(t)}, \quad 0 < t < \pi\]and determine the values of \(t\) for which the curve is concave up.

**EXAMPLE 3 [P.3] | Horizontal and vertical tangents**

Find the points where the curve has horizontal and vertical tangents. It might help also to graph the curve!\[x(\theta) = \cos{(3\theta)}, \quad y(\theta) = 9 \sin{(\theta)}\]