How to Find Derivatives of Functions in Parametric Forms?

Let’s say we have two variables x and y, usually, such variables are related to each other in an implicit or an explicit manner. But in some cases, these variables are related to each other through a third variable. This form is called the parametric form of the equation and the variable is called a parameter. We have x = f(t) and y = g(t) here, t is a parameter.

We will first differentiate x and y with respect to ‘t’ separatly. On differentiating x with respect to ‘t’ we get dx/dt and on differentiating y by ‘t’ we get dy/dt.

Now to the derivative of y with respect to x is given by the formula dy/dx = (dy/dt).(dt/dx). The formula for parametric differentiation is also expressed in the image attached below:

The derivative of such functions is given by chain rule,

Using Chain Rule, dy/dt can be written as

Now, rearranging the terms, we get

, where

Thus, [asand]

Proof of Parametric Differentiation Formula

Since y and x are dependent on ‘t’, then any change in ‘t’ would also cause a change in ‘y’ and ‘x’. Hence, for small change in ‘t’ given as Δt the corresponding changes in x and y are Δx and Δy.

We can write Δy/Δx = (Δy/Δt)/(Δx/Δt)

Taking limit on both sides

lim _Δx→0 Δy/Δx = lim _Δt→0 (Δy/Δt)/lim _Δt→0 (Δx/Δt)

Using the Concept of Limit and Derivatives, we have

dy/dx = (dy/dt)/(dx/dt)

Read More,

• Derivative of Inverse Trigonometric Function
• Chain Rule
• Application of Derivatives

Parametric Differentiation refers to the differentiation of a function in which the dependent and independent variables are equated to a third variable. Derivatives of the functions express the rate of change in the functions. We know how to calculate the derivatives for standard functions. Chain rule, Product rule, and Quotient rule are used to calculate the derivatives of the complex functions which are made up of composition from two or more functions. These functions have two variables that are related to each other in an implicit or explicit manner.

Sometimes we encounter functions in which variables are not related to each other implicitly or explicitly, instead, they are related to each other through a third variable. In this case, we use Parametric Differentiation, which we study in Class 12 for the first time. This article helps in the introduction of the topic to everyone. Here we discussed Parametric Differentiation in detail including methods to find the derivative of a function in parametric form, as well as various other solved examples.