So this is going to be equal to, in the numerator the derivative with respect to t of dy, dx.

And so everywhere we saw a y here, replace that with the first derivative. Now we wanna find theĭerivative with respect to x of the first derivative with respect to x. Of y with respect to t over the derivative of Let me write it this way: When we wanted to find the derivative with respect to x of y, that was equal to derivative What we really wanna do is we wanna find the derivative with respect, Actually, let me just write it down out here a little bit clearer. Here, we wanna find the second derivative of y with respect to x. Of y with respect to t and then divided that by the derivative of x with respect to t. When we wanted to find the derivative of y with respect to x, we found the derivative

Why this is the same thing that we did before, I encourage you to pause the video and think about it. We wanna take the derivative, with respect to T in the numerator, of the first derivative, which I will put in blue now, of dy, dx, all of that over dx, dt. Of the first derivative with respect to t, so So what this is going to be, we wanna find the derivative If you wanna find the rate of change of something with respect to x, you find the rate ofĬhange of that something with respect to t and divide it by the rate of change of x with respect to t. Out the first derivative of y with respect to x in terms of t, now how do we find the second derivative? How do we find the second derivative of y with respect to x now? And I'll give you a hint! We're going to use this same idea. I take away two of those t's, I'm just gonna have a t. This is equal to, so this is gonna be one half, that's three over six, e to the 3t minus 2t. Take the derivative of 2t with respect to t, which is just two, so this is gonna be 6e to the 2t. To the 2t with respect to 2t is going to be e to the 2t and then we're going to The three out front, and so the derivative of e Respect to t is equal to, well, we're gonna have Of x with respect to t? Derivative of x with So it's going to be equal to 3e to the 3t, all of that over, well, what's the derivative And then the derivative of negative one, well, a constant doesn't change no matter what you do to your t, so that's just going to be zero. Of 3t with respect to t is going to be three, so I can say times three like that, or I can put that three out front. Let's see, the derivative of either the 3t with respect to 3t is just e to the 3t. Of y with respect to t? Dy, dt is equal to. And so this is going to be equal to, well, what is the derivative And we've seen thisīefore in other videos, where this is going to be the derivative of y with respect to t over the derivative of So first lets find the first derivative of y with respect to x. Of y with respect to x and the second derivative Possible t's that you can into these functions and then plot the corresponding x and y's for each t, this will plot a curve in the x-y plain. A set of parametric equations where x and y are bothĭefined in terms of t.

Derivative of parametric equations