Derivatives of Functions in Parametric Forms

When x = f(t) and y = g(t), the relation between x and y is said to be in parametric form.

• Write the given equations in parametric form: x = f(t), y = g(t).

• Find \[\frac{dx}{dt}\] and \[\frac{dy}{dt}\] separately.

• Use the formula \[\frac{dy}{dx} = \frac{dy/dt}{dx/dt}\].

• Simplify the answer carefully.

• If required, express the final answer in terms of x and y instead of the parameter.

Find \[\frac{dy}{dx}\], if \[x^{\frac{2}{3}} + y^{\frac{2}{3}} = a^{\frac{2}{3}}\].

Solution: Let \[x = a \cos^3 \theta, y = a \sin^3 \theta\]. Then

Hence, \[x = a \cos^3 \theta, y = a \sin^3 \theta\] is parametric equation of \[x^{\frac{2}{3}} + y^{\frac{2}{3}} = a^{\frac{2}{3}}\]

Now \[\frac{dx}{d\theta} = - 3a \cos^2 \theta \sin \theta \text{ and } \frac{dy}{d\theta} = 3a \sin^2 \theta \cos \theta\]

Therefore \[\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{3a \sin^2 \theta \cos \theta}{-3a \cos^2 \theta \sin \theta} = -\tan \theta = -\sqrt[3]{\frac{y}{x}}\]

• Parametric form means both x and y are written in terms of a third variable.

• The third variable is called the parameter.

• The main formula is:

  \[\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}\]

• This formula is based on the chain rule.

• Always check that \[\frac{dx}{dt} \neq 0\].

• The final answer may remain in terms of the parameter unless the question asks for conversion.