Question

How do you find the second derivative of parametric function?

Answer 1

Hint: In order to determine to determine the second derivative of parametric function or equations , we have to use the chain rule twice and with the help of the result obtained for first derivative , you will see that the second derivative is equal to division of the derivative with respect to $t$ of the first derivative with the derivative of $x$ with respect to t.

Complete step-by-step answer:
To find out the second derivative of a parametric function or equation lets understand what are parametric equations.
Parametric equations are equations in which the dependent variable of derivative i.e. $xandy$ are dependent on some other independent third variable $(t)$ .
 $x=x\left(t\right)$ and $y=y\left(t\right)$
Now the first derivative of dependent variable $y$ with respect to the another dependent variable $x$ comes to be
 $$\begin{gathered} {\displaystyle \frac{dy}{dx}}={\displaystyle \frac{{\displaystyle \frac{dy}{dt}}}{{\displaystyle \frac{dx}{dt}}}} \\ {\displaystyle \frac{dy}{dx}}={\displaystyle \frac{y^{\prime }(t)}{x^{\prime }(t)}} \end{gathered}$$
Here, ${\displaystyle \frac{dx}{dt}}=x^{\prime }(t)$ denotes the derivative of parametric equation $x$ with respect to $t$ and similarly ${\displaystyle \frac{dy}{dt}}=y^{\prime }(t)$ denotes the derivative of parametric equation $y$ with respect to $t$ .
Now to calculate the second derivative of parametric equations, we have to use the chain rule twice.
 $$\begin{gathered} {\displaystyle \frac{d^{2}y}{d{x}^2}}={\displaystyle \frac{d}{dx}}\left({\displaystyle \frac{dy}{dx}}\right)={\displaystyle \frac{d}{dx}}\left({\displaystyle \frac{{\displaystyle \frac{dy}{dt}}}{{\displaystyle \frac{dx}{dt}}}}\right) \\ {\displaystyle \frac{d^{2}y}{d{x}^2}}={\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{{\displaystyle \frac{dy}{dt}}}{{\displaystyle \frac{dx}{dt}}}}\right){\displaystyle \frac{dt}{dx}} \\ {\displaystyle \frac{d^{2}y}{d{x}^2}}={\displaystyle \frac{{\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{{\displaystyle \frac{dy}{dt}}}{{\displaystyle \frac{dx}{dt}}}}\right)}{{\displaystyle \frac{dx}{dt}}}} \end{gathered}$$
Therefore, to find out the second derivative of the parametric function, find out the derivative with respect to $t$ of the first derivative and after that divide it by the derivative of $x$ with respect to t.

Note: 1. Calculus consists of two important concepts one is differentiation and other is integration.
2.What is Differentiation?
It is a method by which we can find the derivative of the function .It is a process through which we can find the instantaneous rate of change in a function based on one of its variables.
Let y = f(x) be a function of x. So the rate of change of $y$ per unit change in $x$ is given by:
 ${\displaystyle \frac{dy}{dx}}$ .
1.Don’t forget to cross-check your answer at least once.
2.Differentiation is basically the inverse of integration.
3. $x$ and $y$ are the dependent variables of the derivative and both $x$ and $y$ are dependent on some independent variable $t$ .