Parametric Differentiation

Parametric Differentiation is the process of differentiating two separate Parametric Equations. Parametric Differentiation is different from the standard Differentiation process as the two separate Equations (y,x) are Functions of a third dependent variable called a parameter. In the standard differentiation method, the Equations are in the Cartesian form where a dependent variable (y) is expressed as a function of another variable (x). For example Cartesian equations can have the form of y=f(x) whereas Parametric Equations have the form of y=f(t) and x=f(t) .

Why do we use parametric differentiation?

Parametric differentiation is used to describe the slope and concavity of parametric curves which are defined by parametric equations. These types of equations often describe curves that inter-lap at several points and are hard to describe with Cartesian equations. The parametric derivatives can then be used to construct the equations of tangents and the normals to curves.

What are the steps for parametric differentiation?

Let's break this down.

Apply the standard chain rule

The first step is to apply the standard Chain Rule as we are still looking for the derivative of dy/dx, but we need to find the correct expression for the given parametric equations. The Chain Rule is as follows:

$\frac{dy}{dx}=\frac{dy}{du}\times \frac{du}{dx}$

Let's look at an example containing the parametric equations of a circle:

Using the chain rule, we now have the following expression :

$\frac{dy}{dx}=\frac{dy}{dt}·\frac{dt}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$

The chain rule can also be expressed as a Ratio of the two derivatives in respect to the common parameter, by rearranging the expression as shown above – provided that the denominator is not zero. This is a trick that simplifies the process, also called the reverse chain rule.

Differentiation of each parametric equation separately

As a second step, we must carry out the differentiation of each equation. We will now proceed with differentiating the x function in respect to t, and then repeat the process for the y function. This will then be used to create the Ratio shown in step 2, to find the parametric derivative dy/dx, by dividing the two derivatives.

Continuing with the first example, the derivatives of x, y Functions with respect to t are as follows:

Create a ratio of the parametric derivatives

The third and last step involves the substitution of each parametric derivative into the ratio expression obtained by the application of the chain rule. Proceeding with the example, we divide the y derivative over the x derivative.

$\frac{dy}{dx}\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{\cos \left(t\right)}{-\sin \left(t\right)}=-cot\left(t\right)$

Finding the equation of a tangent of a curve using parametric differentiation

The tangent of a curve is a straight line that touches the surface of a curve at a specific point$(x_1,y_1)$ as seen in the figure below. Since a tangent is a straight line, the equation of the tangent has the form of y = ax +c where a is the gradient or slope and c is a constant.

The slope of the tangent line is equal to the derivative of the curve. If the point of tangency is known as well as the parametric equations, the equation of the tangent can be found using the parametric differentiation with the following formula, where $x_1,y_1$ are the coordinates of the point of tangency.

$y-y_1=a(x-x_1)$

Example of a tangent of a curve StudySmarter

How to find the equation of a normal to a curve

A normal to a curve is also a straight line that is perpendicular to the tangent of the curve at a specific point$(x_1,y_1)$ as seen in the figure below. Similarly to the equation of a tangent, the equation of the normal can also be found using the derivative at the specified point. However, the slope of the normal must be the negative reciprocal of the derivative of the curve like shown in the formula below where T and N indicate the slope of the tangent and normal respectively.

$Slop{e}_T\times Slop{e}_N=-1$

Example of a normal of a curve Study Smarter