Parametric Derivative Derivation

In summary, the formula for the second derivative in parametric form is incorrect. The correct formula is y'' = \frac{d^2 y}{dx^2} = \frac{d \dot{y'}}{d \dot{x}} = \frac{dy'}{dx}.
  • #1
courtrigrad
1,236
2
How does one conclude that [tex] \frac{d^{2} y}{dx^{2}} = \frac{dy\'/dt}{dx/dt} [/tex]?

Thanks
 
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  • #3
[tex]\frac{d^2 y}{dx^2} = \frac{d \dot{y'}}{d \dot{x}}[/tex]
[tex]= \frac{dy'}{dx} = \frac{dy}{dx^2}[/tex]

Which is clearly incorrect.
 
Last edited:
  • #4
cronxeh said:
Well if you googled "parametric derivative" you would've stumbled on your answer in about 3 seconds:
http://www.mathwords.com/p/parametric_derivative_formulas.htm
And, you will notice that what you wrote was incorrect. What is true is that [tex]\frac{dy}{dx}= \frac{\frac{dy}{dt}}{\frac{dx}{dt}}[/tex] not the second derivative.
That follows from the chain rule.
 
  • #5
I didn't write
[tex]\frac{d^{2}y}{dx^{2}}= \frac{\frac{dy}{dt}}{\frac{dx}{dt}}[/tex]
. I wrote [tex]\frac{d^{2}y}{dx^{2}}= \frac{\frac{dy'}{dt}}{\frac{dx}{dt}}[/tex]
 
  • #6
Yes. Oops.
[tex]\frac{d^2 y}{dx^2} = \frac{d \dot{y'}}{d \dot{x}}[/tex]
[tex]= \frac{dy'}{dx} = y''[/tex]
 
  • #7
Then is y' the derivative with respect to x or t? If with respect to t, then your equation is wrong. If with respect to x, then you are back to the first derivative case.
 
  • #8
[tex]y'' = \frac{d^2 y}{dx^2}[/tex]
 

1. What is a parametric derivative derivation?

A parametric derivative derivation is a mathematical process that involves finding the derivative of a parametric equation. A parametric equation is a set of equations that describe the relationship between two or more variables.

2. How is a parametric derivative derivation different from a regular derivative?

A parametric derivative derivation involves finding the derivative of a parametric equation, which means taking the derivative with respect to one of the variables while holding the other variables constant. This is different from a regular derivative, which involves finding the derivative of a single variable function.

3. What is the purpose of finding a parametric derivative derivation?

The purpose of finding a parametric derivative derivation is to understand the rate of change of a parametric equation. This can be useful in many fields, such as physics, engineering, and economics, where the relationship between variables is described by parametric equations.

4. What are the steps involved in a parametric derivative derivation?

The steps involved in a parametric derivative derivation include identifying the parametric equation, finding the derivatives of each variable with respect to the parameter, and then using the chain rule to simplify the derivatives. The final result will be a single derivative with respect to the parameter.

5. Are there any common mistakes to avoid when performing a parametric derivative derivation?

One common mistake to avoid when performing a parametric derivative derivation is forgetting to use the chain rule. Another mistake is not properly identifying the variables and their relationship in the parametric equation. It's also important to carefully simplify the derivatives and check for any errors before arriving at the final result.

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