Second derivative of multivariate function: area of a rectangle is x*y

[viwaszczenko]

area of a rectangle is x*y
derivative with respect to time is x*dy/dt + y*dx/dt

What is the second derivative with respect to time

 

[Ishuda]

area of a rectangle is x*y
derivative with respect to time is x*dy/dt + y*dx/dt

What is the second derivative with respect to time

What are your thoughts? What have you done so far? Please show us your work even if you feel that it is wrong so we may try to help you. You might also read
http://www.freemathhelp.com/forum/threads/78006-Read-Before-Posting

Hint: Just apply the product rule again.

 

[Deleted member 4993]

area of a rectangle is x*y
derivative with respect to time is x*dy/dt + y*dx/dt

What is the second derivative with respect to time

d/dt(x * dy/dt) = dx/dt * dy/dt + x * d/dt (dy/dt) .....................using product rule of differentiation

and continue.....

 

[viwaszczenko]

d/dt(x * dy/dt) = dx/dt * dy/dt + x * d/dt (dy/dt) .....................using product rule of differentiation

and continue.....

I did that and I want to confirm that the answer is x*d^2y/dt^2 + y*d^2x/dt^2 + (dx/dt)*(dy/dt)

d^2y/dt^2 is the second derivative with respect to time

 

[Ishuda]

I did that and I want to confirm that the answer is x*d^2y/dt^2 + y*d^2x/dt^2 + 2 (dx/dt)*(dy/dt)

d^2y/dt^2 is the second derivative with respect to time

Not quite, see red above:

$\displaystyle \dfrac{d\left(x\, \dfrac{dy}{dt}\, +\, y\, \dfrac{dx}{dt}\right)}{dt}$

= $\displaystyle \dfrac{dx}{dt}\, \dfrac{dy}{dt}\, +\, x\, \dfrac{d^2y}{dt^2}\, +\, \dfrac{dy}{dt}\, \dfrac{dx}{dt}\, +\, y\, \dfrac{d^2x}{dt^2}$

 

[stapel]

d/dt(x * dy/dt) = dx/dt * dy/dt + x * d/dt (dy/dt) .....................using product rule of differentiation

and continue.....

I did that....

In future, kindly please show your work up front, so efforts aren't wasted on repeating what you've already seen. Thank you!

 

[viwaszczenko]

Not quite, see red above:

$\displaystyle \dfrac{d\left(x\, \dfrac{dy}{dt}\, +\, y\, \dfrac{dx}{dt}\right)}{dt}$

= $\displaystyle \dfrac{dx}{dt}\, \dfrac{dy}{dt}\, +\, x\, \dfrac{d^2y}{dt^2}\, +\, \dfrac{dy}{dt}\, \dfrac{dx}{dt}\, +\, y\, \dfrac{d^2x}{dt^2}$

thanks for the clarification