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<a https://www.thestudentroom.co.uk/showthread.php?t=7467335'> Year 12s: what will be the first way you research your future options? >>
<a https://www.thestudentroom.co.uk/showthread.php?t=7467335'> Year 12s: what will be the first way you research your future options? >>

Partial Differentiation

Question is:



Define

u = x^n * y^n
x = cos(at)
y = sin(at)


find du/dt by substituting for x and y

I've started by putting x and y (the cos and sine functions) raised to the power of n into the equation u, do I just proceed to partially differentiate as normal once I've done this?
Any help appreciated

Cheers
Reply 1
Avatar for davros
davros
16
Original post by SwimGood
Question is:



Define

u = x^n * y^n
x = cos(at)
y = sin(at)


find du/dt by substituting for x and y

I've started by putting x and y (the cos and sine functions) raised to the power of n into the equation u, do I just proceed to partially differentiate as normal once I've done this?
Any help appreciated

Cheers


Are you sure you're expected to use partial differentiation for this?

Surely once you've substituted for x and y into the definition of u you can find du/dy using the normal chain rule.
Reply 2
Avatar for SwimGood
SwimGood
OP
8
The question is part of a 'Partial Differentiation' problem sheet, so I assume so.

The next part of the question is solving via implicit differentiation.
Reply 3
Avatar for davros
davros
16
Original post by SwimGood
The question is part of a 'Partial Differentiation' problem sheet, so I assume so.

The next part of the question is solving via implicit differentiation.


OK, so you should have an equivalent rule in your book for working out du/dt in terms of partial derivatives of u w.r.t.x and w.r.t.y together with the derivatives of x and y w.r.t,t
Reply 4
Avatar for ztibor
ztibor
10
Original post by SwimGood
Question is:



Define

u = x^n * y^n
x = cos(at)
y = sin(at)


find du/dt by substituting for x and y

I've started by putting x and y (the cos and sine functions) raised to the power of n into the equation u, do I just proceed to partially differentiate as normal once I've done this?
Any help appreciated

Cheers


Just write down the partial dervatives of u ∂u∂x\frac{\partial u}{\partial x} and ∂u∂y\frac{\partial u}{\partial y}

get the derivatives of x and y w.r.t t dxdt\frac{dx}{dt} and dydt\frac{dy}{dt}

For du/dt
dudt=∂u∂x⋅dxdt\frac{du}{dt}=\frac{\partial u}{\partial x}\cdot \frac{dx}{dt}
or
dudt=∂u∂y⋅dydt\frac{du}{dt}=\frac{\partial u}{\partial y}\cdot \frac{dy}{dt}

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