Total derivative of a partial derivative

In summary, to find \frac{d^2}{dx^2}(\frac{\partial F}{\partial y''}), you first calculate \frac{\partial F}{\partial y''} by regarding y'' as a variable independent of x, y, and y'. Then you differentiate the result with respect to x, taking into account the derivatives of y with respect to x.
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jimmycricket
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Im doing a question on functionals and I have to use the Euler lagrange equation for a single function with a second derivative. My problem is I don't know how to evaluate [itex]\frac{d^2}{dx^2}(\frac{\partial F}{\partial y''})[/itex]. Here y is a function of x, so [itex]y'=\frac{dy}{dx}[/itex].
I know this is probably something I should be able to do by this stage but such is life.
 
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(a) To calculate [itex]\frac{\partial F}{\partial y''}[/itex], treat y'' as a variable independent of x, y, and y':[tex]
\frac{\partial y}{\partial y''} = \frac{\partial y'}{\partial y''} = \frac{\partial x}{\partial y''}= 0 \\
\frac{\partial y''}{\partial y''} = 1.[/tex]
(b) To calculate the second derivative of that with respect to x, give y, y' and y'' their normal meanings: [tex]
\frac{dy''}{dx} = y''' \\ \frac{dy'}{dx} = y'' \\ \frac{dy}{dx} = y' \\ \frac{dx}{dx} = 1.[/tex]
 
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Im a little confused by your answer. I know that [tex]\frac{d}{dx}(\frac{\partial F}{\partial y'})=\frac{\partial^2 F}{\partial y' \partial y}\frac{dy}{dx}+\frac{\partial^2F}{\partial y'^2}\frac{d^2y}{dx^2}+\frac{\partial^2 F}{\partial y'\partial x}[/tex]
But I don't know how to extend this to find [tex]
\frac{d^2}{dx^2}(\frac{\partial F}{\partial y''})[/tex]
 
  • #4
jimmycricket said:
Im a little confused by your answer. I know that [tex]\frac{d}{dx}(\frac{\partial F}{\partial y'})=\frac{\partial^2 F}{\partial y' \partial y}\frac{dy}{dx}+\frac{\partial^2F}{\partial y'^2}\frac{d^2y}{dx^2}+\frac{\partial^2 F}{\partial y'\partial x}[/tex]
But I don't know how to extend this to find [tex]
\frac{d^2}{dx^2}(\frac{\partial F}{\partial y''})[/tex]

It's the same priniciple: to find [itex]\frac{d}{dx}(\frac{\partial F}{\partial y'})[/itex] you first calculate [itex]\frac{\partial F}{\partial y'}[/itex] by regarding y' as a variable independent of [itex]x[/itex] and [itex]y[/itex]. Then you differentiate the result with respect to [itex]x[/itex], regarding y' as the derivative of y with respect to x.

Example: let [itex]F(x,y,y') = x + yy' + y'^2[/itex]. Then [tex]
\frac{\partial F}{\partial y'} = y + 2y' \\
\frac{d}{dx}\left( \frac{\partial F}{\partial y'} \right) = \frac{d}{dx}(y + 2y') = y' + 2y''.
[/tex]

It's the same principle with [itex]\frac{d^2}{dx^2}(\frac{\partial F}{\partial y''})[/itex]. Let [itex]F(x,y,y',y'') = x + y + y'y'' + e^{y''}[/itex]. Then [tex]
\frac{\partial F}{\partial y''} = y' + e^{y''} \\
\frac{d}{dx} \left(\frac{\partial F}{\partial y''} \right) = y'' + y'''e^{y''} \\
\frac{d^2}{dx^2} \left(\frac{\partial F}{\partial y''} \right) = y''' + y^{(4)}e^{y''} + (y''')^2 e^{y''}.[/tex]
 
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1. What is the definition of a total derivative of a partial derivative?

A total derivative of a partial derivative is a measure of the instantaneous rate of change of a function with respect to a particular variable, while holding all other variables constant. It represents the combined effect of both the direct and indirect changes on the function due to a small change in the independent variable.

2. How is the total derivative of a partial derivative calculated?

The total derivative of a partial derivative is calculated by taking the partial derivative of the function with respect to the specific variable, and then multiplying it by the rate of change of that variable with respect to the independent variable.

3. What is the difference between a partial derivative and a total derivative?

A partial derivative measures the rate of change of a function with respect to one specific variable, while holding all other variables constant. A total derivative, on the other hand, takes into account the indirect changes on the function due to a change in the independent variable.

4. Why is the concept of total derivative of a partial derivative important?

The concept of total derivative of a partial derivative is important because it allows us to understand the overall effect of a change in one variable on a function that depends on multiple variables. This is particularly useful in fields such as economics, physics, and engineering where many variables are involved.

5. Can the total derivative of a partial derivative be negative?

Yes, the total derivative of a partial derivative can be negative. This indicates that the function is decreasing with respect to the specific variable, while holding all other variables constant. However, the total derivative itself is a signed quantity, so the sign can change depending on the direction of the change in the independent variable.

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