Use of Derivative Functions in Parametric Form

These equations define the coordinates of points on a curve as functions of parameter t: x = f(t), y = g(t). Here, t is a real number in an interval. The CBSE board exam asks questions related to this equation on a basic level. Representation in the form of a parametric equation is useful for those curves, like curves and ellipses, that are not easily represented as y = f(x) or x = g(y). A unit circle is expressed in a parametric equation as shown below: x=cost, y=sint. Here, t ranges over [0,2π]. A parabola $y=x^2$ can be parametrically represented as: x=t, $y=t^2$

 Let us now take a look at the first derivative and second derivative:

1. First Derivative $\left(\frac{dy}{dx}\right)$

The first derivative of y with respect to x for a curve that is defined parametrically is given by $\frac{dy}{dx}=\frac{\frac{d\mathrm{y/dt}}{}}{\frac{d\mathrm{x/dt}}{}}$

The formula derived from chain rule states that $\frac{dy}{dt}=\frac{dy}{dx}\cdot \frac{dx}{dt}$

After rearranging the above equation, we will get the derivative in a parametric form. Questions based on this will be asked in IIT JAM entrance exam and JEE Main exam.

The derivative $\left(\frac{dy}{dx}\right)$ shows the slope of the tangent line to the curve at any point.

2. Second Derivative $\left(\frac{d^{2}y}{d{x}^2}\right)$

Second derivative is the derivative of first derivative w.r.t. x. This derivative determines both concavity and curvature of a curve. $\frac{d^{2}y}{d{x}^2}=\frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{d\mathrm{x/dt}}{}}$

By using the quotient rule, it will be:

$\frac{d^{2}y}{d{x}^2}=\frac{\frac{d}{dt}\left(\frac{dy}{dt}\right)\cdot \frac{dx}{dt}-\frac{dy}{dt}\cdot \frac{d^{2}x}{d{t}^2}}{{\left(\frac{dx}{dt}\right)}^3}$

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Let us understand the applications of derivatives in parametric form:

• Finding Slope of Tangent Lines: The first derivative gives slope of tangent line to a parametric curve at any point. This is important for understanding the direction and steepness of curve at any point.
• Determining concavity: The second derivative $\frac{d^{2}y}{d{x}^2}$ helps in determining the concavity of the curve. If    $\frac{ d^2 y }{ d x^2 }$ >0 , the curve is concave upwards. On the other hand, if $\frac{d^2 y }{ d x^2 }$<0, the curve will concave downwards.
• Identifying Tangents: Vertical tangents occur when $\frac{{\mathrm{dx}}^{}}{d{t}^{}}$= 0 and horizontal tangents occur when  $\frac{{\mathrm{dy}}^{}}{d{t}^{}}$= 0.
• Analyzing Motion: In Physics, parametric equations describe the position of an object as a function of time t. Derivatives $\frac{{\mathrm{dx}}^{}}{d{t}^{}}$and $\frac{{\mathrm{dy}}^{}}{d{t}^{}}$represent the velocity component of objects. 

Let us take a look at the mistakes that must be avoided when using the parametric form. NEET exam aspirants must keep these pointers in mind.

• Forgetting to divide by $\frac{dx}{dt}$: The derivative $\frac{dy}{dx}$ is the ratio $\frac{\frac{d\mathrm{y/dt}}{}}{\frac{d\mathrm{x/dt.}}{}}$, not just $\frac{dy}{dt}$.
• Incorrect differentiation: Ensure proper application of differentiation rules (chain rule, product rule, quotient rule) when computing $\frac{dx}{dt}$ and $\frac{dy}{dt}$.
• Ignoring special cases: Always check for $\frac{dx}{dt}=0$ or $\frac{dy}{dt}=0$ to identify vertical or horizontal tangents.