Both cases are using the chain rule. The chain rule states
In the first problem, let u = 2t, in the secont problem, let u = sec(3t). TO solve that you will have to use the chain rule again, letting your y be sec(3t) and your u to be = 3t.
So my calculus book gives me
I have to differentiate
So, for the first one, I would take to be and get
but it says it should be
I don't understand how this works.
Also for the second one they get
and using the principle that I learned from the first one it makes sense except that the secant should not be squared.
What? So you let u be 2t. Then you have . Now, y=sin(u), u=2t.
Hence, , where I have resubstituted the u=2t at the end. Now do the same for your second equation, only now you let y= u^2, u = sec(3t), and to work out what the derivitive of sec(3t) is you use the chain rule again, letting u=3t, y=sec(u).
We need the derivative of the whole and the derivative of the inside. To better see this, let us rewrite
So our "outside" is the whole bracket, where we use power rule, and the "inside" is sec3t.
So, the derivative of the whole is
But due to chain rule, we need the derivative of the inside.
But of course, we can notice that we have another function "3t" that also has a derivative, so we must multiply by this as well.
So, you have , with . Now, you want to defrentiate both sides by x.
You have to remember what it is you are defrentiating with respect to. For the RHS, you defrentiated with respect to z, for the LHS with respect to x, so of course they are not equal. They are related however, by
Hm...I was trying implicit differentiation starting with , but I suppose that wouldn't work unless one was a function of the other....and so when you multiply the right side by you get , and when , would be .
Also, just a tip, it's spelled "differentiating" lol