Partial derivatives with Wave Function

[abelanger]

Homework Statement

Knowing: y(x,t) = Acos(kx-ωt)
Find the partial derivatives of:
1) dy/dt
2) dy/dx
3) d^2y/dt^2
4) d^2y/dx^2

Homework Equations

The Attempt at a Solution

These are the answers the actual answers:
1) dy/dt = ωAsin(kx-ωt) = v(x,t) of a particle
2) dy/dx = -kAsin(kx-ωt)
3) d^2y/dt^2 = -(ω^2)Acos(kx-ωt) = a(x,t) of a particle
4) d^2y/dx^2 = -(k^2)Acos(ks-wt)

now here are my questions:
1) how come when I do the partial derivative of y with respect to t, kx becomes the constant and vice versa with dy/dx, how come ωt becomes the constant? is it because of implicit differentiation?
2) What does it give me to find: dy/dx? the slope? also what about d^2y/dx^2?

Thanks for the Help!

 

[klondike]

also what about d^2y/dx^2?

Thanks for the Help!

$$\dfrac{\partial ^{2}y}{\partial x^{2}}=\dfrac {1}{c^2}\dfrac{\partial ^{2}y}{\partial t^{2}}$$

 

[abelanger]

$$\dfrac{\partial ^{2}y}{\partial x^{2}}=\dfrac {1}{c^2}\dfrac{\partial ^{2}y}{\partial t^{2}}$$

Ooooh.. Interesting!

Thanks