Derivative Of a function with respect to a different variable.

Question

Today in a standard calculus course in high school,i notice that our sir wrote : $$\frac{d}{dx} \cos(2t)=\frac{d}{dt}(\cos2t).\frac{dx}{dt}$$ But i am supposed to write as : say $u=\cos(2t)$ Then $$\frac{du}{dx}=\frac{d}{dt}(\cos2t).\frac{dt}{dx}$$ That's what the chain rule says : Please explain which of these is right and explain.

Answer 1

Yes, yours is correct clearly. $$\frac{du}{dx} = \frac{du}{dt}\cdot\frac{dt}{dx}$$ This way, the $dt$’s balance, leaving you with $\frac{du}{dx}$.