How to Find the Third Derivative of Parametric Curves

TL;DR

To find the third derivative of a parametric function, first calculate the first derivative by determining dy/dx using the derivatives dx/dt and dy/dt. Then, compute the second derivative using d^2y/dx^2, and finally find the third derivative by differentiating the second derivative with respect to t and dividing by dx/dt.

Transcript

in this video we're going to talk about how to find the third derivative of a parametric function so let's say that x is equal to t cubed minus three and y is equal to t to the fifth power minus t to the fourth power how can we find the third derivative well we need to find the first derivative the second and then the third so let's start with the ... Read More

Key Insights

• ❓ The process of finding the third derivative of a parametric function involves finding the first, second, and then third derivative.
• ✊ The power rule is crucial in finding the first and second derivatives of the parametric function.
• 🍞 The formula for finding the second derivative in parametric form is d^2y/dx^2 = (dy/dt) / (dx/dt).

Questions & Answers

Q: How do you find the first derivative of a parametric function?

To find the first derivative, you need to find dx/dt and dy/dt using the power rule. dx/dt is obtained by differentiating the x equation, and dy/dt is obtained by differentiating the y equation.

Q: What is the formula for finding the second derivative in parametric form?

The formula is d^2y/dx^2 = (d/dt(dy/dx)) / (d/dt(dx/dt)). You substitute the first derivative dy/dx and dx/dt into this formula to find the second derivative.

Q: How do you find the third derivative of a parametric function?

To find the third derivative, you differentiate the second derivative with respect to t and divide it by dx/dt. This will give you the third derivative of the parametric function.

Q: Can you provide an example of finding the third derivative?

Let's say x = t^3 and y = 2t^6 + 3t^4. By following the steps outlined in the video, you can find the first, second, and third derivatives to get the final answer for the third derivative.

Summary & Key Takeaways

• The video discusses the process of finding the first derivative of a parametric function by finding dx/dt and dy/dt using the power rule.

• The second derivative is then found by applying the formula for d^2y/dx^2, using the first derivative and dx/dt.

• Finally, the third derivative is found by differentiating the second derivative with respect to t and dividing it by dx/dt.