YAHomeworkT - Derivatives and Mechanics

[QueHuong]

In calc, I've always been taught that if y = x, then dy/dx = 1.

But now in Mechanics, it's actually 1dx (or 1/dx?). In calc, we ignored that differentiated x and just had it equal to 1, but now I have to explicity state the differentiated x.

To illustrate more clearly, here's a practice problem for my exam that I have to derive:

(d/dt)(1+(Cos(3THETA))^2)^(1/2)

I got

.5(1+(cos(3THETA))^2)^-0.5 (2cos(3THETA))(-sin(3THETA))(3)

In calc, that would have been fine, but in Mech, it's wrong b/c I didn't write down the derivative of Theta:

.5(1+(cos(3THETA))^2)^-0.5 (2cos(3THETA))(-sin(3THETA))(3THETA DOT) (with Theta dot meaning the derivative of theta).

But what's the significance of Theta Dot and what would be an example where theta dot becomes important?

 

[Martin]

putting the original function symbol (theta, x etc) with a dot on top is just another way of writing the derivative. As you probably know 2 dots on top means second derivative.

I think its to prepare you for differential equations, since there you use this notation.

 

[dopcombo]

its because when u learnt dy/dx =1, you assumed that there was only a relationship between y and x and nothing else.
it wasn't say for example, y(t) and x(t)

in this case, maybe your theta can change wrt time, and hence when u differentiate it, it must be considered a function, and it must be differentiated as well.

 

[RossGr]

The (dot) is Common notation for a derivative with respect to time.

So in your problem Theta is dependent on time, to neglect that term would be wrong in your calc class as well.

Recall the chain rule? (f(g(x))' = f'(g(x)) * g'(x). Here I use the prime to denote a derivative wrt to x.
To get the correct derivative to your given problem you must use the chain rule.

You should have seen similar things while learning implicit differentiation.