Hello, I am encountering some major confusion. When taking just garden variety f(x)=y derivatives of the form dy/dx, I don't encounter any problems. But recently I started taking derivatives of parametric equations, or switching things up using polar equations and I realized perhaps I'm not so solid on these things as I once thought. Is there any way someone could tell me if I'm right about a few problems? So say we have: 2x2+3x if we go (d/dx)(2x2+3x) the answer is just (4x+3) Now what if x is actually a function of another variable, say, t for instance. Say we know in reality, that x=2t. Now is (d/dx)(2x2+3x) still (4x+3)? after all, we are only asking about the differential in terms of x Now (d/dt)(2x2+3x) = (d/dx)(dx/dt)(2x2+3x) = (4x+3)(2) Right? Ok so now what if we have y=2x2 x=4t (dy/dx)=4x (d/dx)(2x2)=4x Also, (dx/dt)=4, which is the same as writing (d/dt)(4t)=4 Now, we could also write: (dy/dx)=(dy/dt)/(dt/dx) If we plug in (4t) for x, we get y=2(4t)2=32t2 So dy/dt=64t We know (dx/dt)=4 --> (dt/dx)=1/4 (since this denominator never goes to 0) so (dy/dx) = (dy/dt)*(dt/dx)=64t*(1/4) = 16t From the equation x=4t given above we know, t = (1/4)x so (dy/dx)= 16t= 16(x/4) = 4x Is this all right? ~~~~~~~~~~~~~~~~ Another confusing question for me: say we have x=3t Would d/dx(sinx)=cosx Or would it be d/dx(sinx)=(cosx)(x')=(cosx)(dx/dt)=(cosx)(3) When I see d/dx, I always think that just means, "take the derivative of this in terms of x only". What if we know x is a function of another variable though?