Derivatives of Functions in Parametric Forms

When x and y are expressed separately as functions of the same third variable t, i.e. \[ x = f(t), \qquad y = g(t), \] the equations are called parametric equations, and t is called the parameter.

The parameter may also be denoted by \[\theta, u,\] etc.

Important Condition

The formula \[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \] is directly applicable when

\[ \boxed{\frac{dx}{dt} \neq 0.} \]

If \[ \frac{dx}{dt} = 0, \] the point must be examined separately. It may correspond to a point where the tangent is vertical.

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

If \[ x = a\cos t, \qquad y = a\sin t, \] find \[ \frac{dy}{dx}. \]

Solution:

Differentiating x with respect to t,

\[ \frac{dx}{dt} = -a\sin t. \]

Differentiating y with respect to t,

\[ \frac{dy}{dt} = a\cos t. \]

Therefore,

\[ \frac{dy}{dx} = \frac{a\cos t}{-a\sin t} \]

\[ \boxed{\frac{dy}{dx} = -\cot t} \]

• 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.